1
Decomposition of Graphs into Paths
Fábio Botler and Yoshiko Wakabayashi
E-mail: {fbotler,yw}@ime.usp.br
Abstract—A decomposition of a graph G is a set D = {H1 , · · · , Hk } of pairwise edge-disjoint subgraphs of G that cover the set of
edges of G. If Hi is isomorphic to a fixed graph H , for 1 ≤ i ≤ k, then we say that D is an H -decomposition of G. In this work, we
study the case where H is a path of fixed length. For that, we first decompose the given graph into trails, and then we use a
disentangling lemma, that allows us to transform this decomposition into one consisting only of paths. With this approach, we tackle
three conjectures on H -decomposition of graphs and obtain results for the case H = P` is the path of length `. Two of these results
solve weakenings of a conjecture of Kouider and Lonc (1999) and a conjecture of Favaron, Genest and Kouider (2010), both for regular
graphs. We prove that, for every positive integer `, (i) there is a positive integer m0 such that, if G is a 2m`-regular graph with m ≥ m0 ,
then G admits a P` -decomposition; (ii) if ` is odd, there is a positive integer m0 such that, if G is an m`-regular graph with m ≥ m0 and
containing an m-factor, then G admits a P` -decomposition. The third result concerns highly edge-connected graphs: there is a positive
integer k` such that if G is a k` -edge-connected graph whose number of edges is divisible by `, then G admits a P` -decomposition.
This result verifies for paths the Decomposition Conjecture of Barát and Thomassen (2006), on trees. This work is an extended
abstract of the Ph.D. thesis of the first author, written under the supervision of the second author.
Index Terms—graph, path decomposition, highly edge-connected, regular graph.
F
1
I NTRODUCTION
A decomposition D = {H1 , . . . , Hk } of a graph G is a set of
edge-disjoint subgraphs of G that cover the edge set of G.
An implication of Hall’s Theorem (1935) states that bipartite
regular graphs admit a decomposition into perfect matchings. A consequence of this fact, already known by Petersen
in 1891, is that even regular graphs admit decompositions
into 2-factors. Here, we say that a graph G is even (resp. odd)
if every vertex of G has even degree (resp. odd degree).
Since then, many results on decompositions have appeared in the literature. For example, Pyber [41] proved that
every simple graph can be decomposed into at most four
odd graphs. Jünger, Reinelt e Pulleyblank [30] considered
decompositions into connected subgraphs with k edges
(and at most one subgraph with less than k edges), called
k -partitions. They proved that every k -edge-connected graph
admits a (k + 1)-partition, for k = 1, 2, 3, and every 4-edgeconnected graph admits an s-partition, for every positive
integer s. They also conjectured that 2-edge-connected planar graphs admit 3-partitions composed only by trails.
In this work we are interested in decompositions D =
{H1 , . . . , Hk } of a graph G in which Hi is isomorphic to a
fixed graph H , for every 1 ≤ i ≤ k . Such a decomposition
is called an H -decomposition. As noted by Häggkvist [25], a
natural question is, given two graphs G and H , to decide
if G admits an H -decomposition.
“Given two graphs G and H and an inquisitive mind we may ask whether or not G
is the edge-disjoint union of copies of H .”
(HÄGGKVIST, 1989)
This research has been partially supported by CNPq Projects (Proc.
477203/2012-4 and 456792/2014-7), Fapesp Project (Proc. 2013/03447-6) and
MaCLinC Project of Numec/USP, Brazil. F. Botler is supported by Fapesp
(Proc. 2014/01460-8 and 2011/08033-0), and Y. Wakabayashi is partially
supported by CNPq Grant (Proc. 303987/2010-3)
An obvious necessary condition is that |E(G)| is divisible
by |E(H)|. Jünger, Reinelt and Pulleyblank’s result above
implies that if G is connected and has an even number of
edges, then G admits a P2 -decomposition, where P2 is the
path with two edges. On the other hand, Dor and Tarsi [17]
proved that when H is connected and has at least 3 edges,
the problem of deciding if G admits an H -decomposition is
NP-complete. Thus, it is natural to search for sufficient conditions to obtain such decompositions. For decompositions
into triangles, for example, Nash-Williams [38] conjectured
that minimum degree 3n/4 is sufficient, i.e., if G is a graph
on n vertices and minimum degree δ(G) ≥ 3n/4 (and
|E(G)| is divisible by 3), then G admits a K3 -decomposition.
To approach Nash-Williams’ Conjecture, Barber, Kühn, Lo
and Osthus [4] proved that even graphs with minimum
degree δ(G) ≥ 9n/10 + o(n) and such that |E(G)| is
divisible by 3 admit K3 -decompositions.
In the case where H = T is a tree, Barát and
Thomassen [3] conjectured that high edge-connectivity is
sufficient, i.e., they proposed the following conjecture.
Conjecture 1.1 (Barát–Thomassen, 2006). For each tree T ,
there is a positive integer kT such that, if G is a kT -edgeconnected graph and |E(G)| is divisible by |E(T )|, then G admits
a T -decomposition.
Barát and Thomassen’s Conjecture, known as the Decomposition Conjecture, is the main subject of this work.
Thomassen [46], [47], [48], [49], [50] proved this conjecture
for stars, some bistars, paths of length 3, and paths of
length 2k , for every positive integer k . In this work, we
prove that this conjecture holds for paths of every fixed
length.
To prove Barát and Thomassen’s Conjecture (for paths),
we first study T -decompositions of regular graphs. In 1964,
Ringel [42] conjectured that the complete graph K2`+1
admits a T -decomposition for any tree T with ` edges.
2
Ringel’s Conjecture is commonly confused with the Graceful
Tree Conjecture that says that every tree T with n vertices
admits a labeling f : V (T ) → {0, . . . , n − 1} such that
{1, . . . , n − 1} ⊆ {|f (x) − f (y)| : xy ∈ E(T )}. since the
Graceful Tree Conjecture implies Ringel’s Conjecture [43],
Ringel’s Conjecture holds for many classes of trees, such as
stars, paths, bistars, carterpillars, and lobsters (see [18], [24]).
Häggkvist [25] generalized Ringel’s Conjecture for regular
graphs.
Conjecture 1.2 (Graham–Häggkvist, 1989). For each tree T
with ` edges, if G is a 2`-regular graph, then G admits a T decomposition.
Häggkvist [25] also proved that Conjecture 1.2 holds
when the girth of G is at least the diameter of T . For
more results on decompositions of regular graphs into trees,
see [20], [22], [27], [29], [44]. In the particular case where
H = P` is the path with ` edges and G is a regular graph,
Kouider and Lonc [32] improved Häggkvist’s result, proving that a 2`-regular graph G with girth g ≥ (`+3)/2 admits
a P` -decomposition D such that every vertex of G is the endvertex of exactly two paths of D. They also conjectured that
this fact must hold for every 2`-regular graph, i.e., that every
2`-regular graph G admits a P` -decomposition D such that
each vertex of G is the end-vertex of exactly two paths of D.
In Section 4, we prove a weakening of Kouider and Lonc’s
Conjecture. We prove that, for each positive integers ` and g
such that g ≥ 3, there is m0 = m0 (`, g) such that, if G is a
2m`-regular graph with m ≥ m0 and girth at least g , then G
admits a P` -decomposition D such that each vertex of G is
the end-vertex of exactly 2m paths of D.
Another result related to the result stated above is due to
Kotzig [31], that proved that a 3-regular graph G admits a
P3 -decomposition if and only if G contains a perfect matching. Favaron, Genest, and Kouider [21] conjectured that this
result may be generalized, i.e., that odd `-regular graphs
that contain perfect matchings admit P` -decompositions. In
this case, the degree of the vertices of the graph is decreased
by one-half, but a perfect matching is required. In Section 4,
we prove a weakening of Favaron, Genest, e Kouider’s
Conjecture. We prove that, for each positive integers ` and g
such that ` is odd and g ≥ 3, there is m0 = m0 (`, g) such
that, if G is an m`-regular graph with m ≥ m0 , girth at
least g , and containing an m-factor, then G admits a P` decomposition D such that every vertex of G is the endvertex of exactly m paths of D. Finally, as suggested by
Häggkvist, we conjecture that this fact may hold in a more
general way, with m0 = 1.
“It is a trend in modern mathematics that if you
can not solve a particular problem you can at least
formulate a more general one, thus increasing the
amount of frustration in the world and keeping
your collegues on their toes.” (HÄGGKVIST, 1989)
The main tool developed in this work is a disentangling
lemma (see Section 3), that showed to be useful to deal
with Graham and Häggkvist’s, and Favaron, Genest, and
Kouider’s Conjectures in Section 4, as well as Barát and
Thomassen’s Conjecture in Section 5. Another important
concept is the one of complete decomposition, first introduced
in [12], and that will be used frequently in this work.
Roughly speaking, here the completeness (of a decomposition) represents a special set of properties of a decomposition. Throughout our proofs, we develop an algorithm that
receives a complete decomposition D and returns a “better”
complete decomposition D0 (in the sense that D0 is closer
to the desired decomposition). Therefore, given a graph G
we first construct a decomposition D of G, prove that D is
a complete decomposition, and use the algorithm to obtain
the desired decomposition.
In Section 2, we present briefly the concepts about
graphs that are more important in this work. In the following sections, we reorganize the results in [10], [11] as
follows. In Section 3, we present a disentangling lemma
(Lemma 3.2), that allows us, under some conditions, to
switch edges between elements of a decomposition of a
graph G into trails and obtain a new decomposition of G
into paths. In Section 4, we use Lemma 3.2 to obtain decompositions into paths of fixed length of a large family of regular graphs. In Section 5, we generalize part of the technique
developed by Thomassen that deals with decompositions of
highly edge-connected graphs. This technique give us the
structure needed to apply some of the lemmas obtained in
Section 4, and prove Barát and Thomassen’s Conjecture for
paths of any fixed length.
2
N OTATION AND BASIC CONCEPTS
The basic terminology and the notation used in this work
are standard [7], [16]. A multigraph is a pair G = (V, E). The
sets V and E , also denoted by V (G) and E(G), are, respectively, the sets of vertices and of edges of G. We say that a
multigraph is simple or, simply, a graph if it does not contain
parallel edges. All graphs and multigraphs considered here
are finite and have no loops. In this work we use mainly
graphs, except in Section 5, more specifically, in the proof of
Lemma 5.17. Thus, the following definitions are presented
for graphs, but can easily be extended for multigraphs.
We denote by dG (v) the degree of a vertex v ∈ V
and, when G is clear from context, we write only d(v).
If d(v) = 0, then we say that v is an isolated vertex in G.
If d(v) = r for every vertex v of G, then we say that G
is r-regular. We say that a graph H is a subgraph of G if
V (H) ⊆ V (G) and E(H) ⊆ E(G), and we use frequently
the arithmetic operators + and − in the following way. If F
is a set of edges, and e is an edge, we denote by F + e
and F − e the sets F ∪ {e} and F \ {e}, respectively. If H is
a subgraph of G, and e is an edge of G, then we denote by
H + e the graph obtained from (V (H), E(H) + e) by adding
the vertices of e that are not present in V (H); and by H − e
the graph obtained from (V (H), E(H)−e) by removing the
vertices incident to e that become isolated. Given F ⊆ E ,
we denote by G[F ] the subgraph of G induced by F , i.e., the
graph obtained from V (G), F by removing the isolated
vertices. We also denote by dF (v) the number of edges of F
incident to v or, equivalently, the degree of v in G[F ].
Let G = (V, E) be a graph. A trail in G is a subgraph
of G that admits a sequence of vertices T = v0 v1 · · · v` such
that vi vi+1 ∈ E , for 0 ≤ i ≤ ` − 1, and vi vi+1 6= vj vj+1 ,
for 0 ≤ i < j ≤ ` − 1. It is also convenient to refer to a
trail T = v0 v1 · · · v` as the subgraph of G induced by the
edges vi vi+1 for i = 0, . . . , ` − 1. The length of a trail is
3
its number of edges. We also say that a trail of length `
is an `-trail. A path in G is a trail that admits a sequence
P = v0 v1 · · · v` such that vi 6= vj , for 0 ≤ i < j ≤ `.
We also say that v0 and v` are the end-vertices of P , and
that P joins v0 to v` . A cycle is a trail that admits a sequence
C = v0 v1 · · · v`−1 v0 such that vi 6= vj , for 0 ≤ i < j ≤ ` − 1.
The path of length `, also called the `-path, is denoted by P`
(this notation is not standard). Given two vertices, u and v
in G, the distance between u and v is the length of a shortest
path joining u and v in G; the diameter of G is the largest
distance between two vertices of G; and the girth of G is the
length of a shortest cycle in G.
An orientation O of a subset F ⊆ E , is an assignment
of a direction (from a vertex to the other) to each edge
of F . If an edge e = uv in F is directed from u to v , we
say that e leaves u and enters v . Given a vertex v of G, we
−
denote by d+
O (v) (resp. dO (v)) the number of edges of F that
leave (resp. enter) v with respect to O. In this work we use
the term Eulerian graph as a synonym of even graph, i.e., an
Eulerian graph is a graph that contains only vertices of even
degree. In other words, here we allow an Eulerian graph to
be disconnected, and avoid using the term even to refer to
graphs. An Eulerian orientation of an Eulerian graph G is an
−
orientation O of E(G) such that d+
O (v) = dO (v) for every
vertex v in V . Moreover, we say that a subset F ⊆ E is
Eulerian if G[F ] is Eulerian. We denote by G = (A, B; E) a
bipartite graph G with bipartition A, B of its set of vertices.
We say that a set {H1 , . . . , Hk } of subgraphs of G is a deSk
composition of G if i=1 E(Hi ) = E and E(Hi ) ∩ E(Hj ) = ∅
for every 1 ≤ i < j ≤ k . Let H be a family of graphs. An Hdecomposition D of G is a decomposition of G such that each
element of D is isomorphic to an element of H. Moreover, if
H = {H}, then we say that D is an H -decomposition.
We say that a subgraph H of G is spanning or a factor
of G if V (H) = V (G). A factorization of a graph G is a
decomposition of G into factors. A k -factor of G is a k regular factor of G, and a k -factorization of G is a decomposition of G into k -factors. The following result, Petersen’s
2-Factorization Theorem [40], is essential for the induction
steps in our proofs.
Theorem 2.1 (Petersen, 1891). If G is a 2`-regular graph,
then G admits a 2-factorization.
Let G be a connected graph. The edge-connectivity of G is
the smallest integer k such that there is a set F ⊆ E(G) with
|F | = k and such that G − F is disconnected. If G has edgeconnectivity k , then we say that G is k -edge-connected. The
celebrated Menger’s Theorem [35] says that given a positive
integer k , a graph G is k -edge-connected if and only if for
every pair of vertices u and v there are at least k pairwise
edge-disjoint paths joining u to v in G.
The remaining concepts that we need will be presented
in the forthcoming sections. In Section 3 we define tracking,
used to specify the order in which the vertices of a trail
is visited. In Section 5 we define fractional factorizations and
bifactorizations that extend the concept of factorization, introduced to deal with highly edge-connected bipartite graphs.
3
A DISENTANGLING LEMMA
To decompose a graph G into copies of a fixed graph H , we
use the following strategy: we obtain first a decomposition
of G into elements similar to H ; afterwards, if needed,
we perform edge-switchings and obtain copies of H . For
example, to decompose a graph into paths of length `, we
can find first a decomposition D into trails of length `. If
the trails in D have no cycles, then D is a decomposition
into paths. This occurs, for example, in the case of graphs
with large girth, more specifically, girth larger than `. If in
D there are trails that are not paths, we use disentangling
techniques, this is, an edge-switching step (between the
elements of D), in order to find a new decomposition D0
with less elements that contain cycles.
Heinrich, Liu, e Yu [26] show how to decompose 3mregular graphs that contain an m-factor into paths of
length 3. For that, they construct a decomposition of the
graph into trails with 3 edges, i.e., 3-paths and triangles, and
show that if in this decomposition there is any triangle X ,
then there is another element Y such that X ∪ Y can be decomposed into two paths of length 3. In [12], we decompose
a 5-regular graph, that contain a perfect matching and has
no triangles, into trails of length 5 that may contain a C4 (see
Figure 1). Then we show that if we switch edges between the
elements of this decomposition in a specific order, we obtain
a decomposition into 5-paths. In [32], Kouider and Lonc
decompose 2`-regular graphs with girth at least (` + 3)/2
into trails of length `, and show that for each trail X0 that
is not a path there is a sequence X0 X1 · · · Xk of elements
of this decomposition such that it is possible two switch an
edge of Xi with Xi+1 (i = 0, . . . , k − 1) and obtain only
paths of length `.
Fig. 1: Trails of length 5 in triangle-free graphs.
In this section we present a disentangling lemma that
we developed [10] for paths of any fixed length. More
specifically, we prove a result, Lemma 3.2, that guarantees
that, given a decomposition of a graph into certain special
trails, it is possible to switch edges between the elements
of this decomposition and construct a decomposition with
more paths (that preserves some properties). Our technique
is similar to the technique introduced by Kouider and
Lonc [32], however here we make use of a sufficiently high
minimum degree to compensate the requirement of a large
girth.
3.1
Trails, trackings, and augmenting sequences
We recall that a trail is a graph T for which there is a
sequence B = x0 · · · x` of its vertices (possibly with repetitions) such that E(T ) = {xi xi+1 : 0 ≤ i ≤ ` − 1} and
xi xi+1 6= xj xj+1 , for i 6= j . Such a sequence B of vertices is
called a tracking of T , and we say that T is the trail induced
by the tracking B . We say that the vertices x0 and x` are the
end-vertices of B . Note that a path admits only two possible
trackings, while a cycle of length ` admits 2` trackings.
Given a tracking B = x0 · · · x` , we denote by B − the
tracking x` · · · x0 , and, for ease of notation, we denote
by V (B) and E(B) the sets {x0 , . . . , x` } of vertices and
{xi xi+1 : 0 ≤ i ≤ ` − 1} of edges of B , respectively.
4
Moreover, we denote by B̄ the trail V (B), E(B) . It is
convenient to say that a tracking B = x0 · · · x` traverses the
vertices x0 , . . . , x` and the edges x0 x1 , . . . , x`−1 x` (in these
orders), and that x0 x1 is the initial edge of B and x`−1 x`
is the final edge of B , or that B starts with x0 x1 and
ends with x`−1 x` . A set B of edge-disjoint trackings of a
graph G is a tracking
decomposition (or decomposition into
S
trackings) of G if B∈B E(B) = E(G). Note that the set
B̄ = {B̄ : B ∈ B}, called decomposition subjacent to B , is
a decomposition of G into trails. We also note that if Bi
and Bj are trackings of a tracking decomposition B such
that E(Bi ) ∩ E(Bj ) 6= ∅, then Bi = Bj and, consequently,
B̄i = B̄j (i.e., Bi and Bj induce the same trail).
We say that a tracking x0 x1 · · · x` is a vanilla tracking if
the tracking x1 · · · x`−1 induces a path in G, and we say
that a trail T is a vanilla trail if there is a vanilla tracking
of T (see Figure 2). If a vanilla tracking (resp. vanilla trail)
contains ` edges, then we say that it is an `-vanilla tracking
(resp. `-vanilla trail). Here we fix the central object of this
work: we say that a tracking decomposition B is an `decomposition if every element of B is an `-vanilla tracking.
Note that we do not carry the terms “tracking” or “vanilla”
in `-decomposition, i.e., in this text, every element of an `decomposition is a vanilla tracking. If the decomposition
subjacent to B is a decomposition into paths of G, then we
say that B is an `-decomposition into paths. We may omit the
length `, when it is clear from context.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2: Examples of vanilla trails. The red edges are the initial
and final edges.
Suppose there is a set F of edges of a graph G such that
G − F admits a decomposition D0 into paths. Moreover,
suppose we can extend each path in D0 with one edge
of F at each of its end-vertices, and such that each edge
of F is used exactly once. In this case, each path of D0
becomes a vanilla trail. Moreover, we can obtain a tracking
decomposition B of G. Now fix a vertex v of G. The
number h(v) of trackings of B that end immediately after
traversing v is exactly the number of trails of D0 that have v
as an end-vertex. When h(v) is large with respect to the
maximum length of a tracking of B some properties arise.
The following definition presents such a property that is
fundamental in this subsection.
Definition 3.1 (Feasibility). Let ` be a positive integer. Let G be
a graph and B an `-decomposition of G. We say that B is feasible
if for every v ∈ V (G) the following holds: if T is an `-trail of G
(not necessarily admitting a tracking in B ) and v is contained in
a cycle of T , then there an element B in B such that B contains
an edge vu where u is an end-vertex of B and u ∈
/ V (T ).
Lemma 4.6 presents a sufficient condition for the feasibility of `-decompositions of graphs with restrictions on
the girth. Lemma 4.15 deals with the special case where the
graph is bipartite.
For each vanilla tracking B in G, let τ (B) be the
number of end-vertices of B with degree greater than 1,
and let B
Pa feasible `-decomposition of G that minimizes
τ (B) = B∈B τ (B) in Figure 2a, we have τ (B) = 1, and
in Figures 2b-f, we have τ (B) = 2 . If τ (B) = 0, then B is
an `-decomposition into paths. Thus, suppose that τ (B) > 0.
Since τ (B) > 0, there is a vanilla tracking B0 in B that is not
a path. Let x be an end-vertex of B0 with degree greater
than 1 in B̄0 , and let C be a cycle in B̄0 that contains x.
Consider a neighbor v of x in C . Since B is feasible, there
is an element B1 in B that contains the edge vu, such that
u∈
/ V (B0 ) and u is an end-vertex of B1 . Now, let B00 and
0
B1 such that B̄00 = B̄0 − vx + vu, B̄10 = B̄1 − vu + vx, and let
B 0 = B − B0 − B1 + B00 + B10 . We have τ (B00 ) = τ (B0 ) − 1.
Suppose that the decomposition obtained B 0 is feasible.
If τ (B10 ) ≤ τ (B1 ), P
then B 0 is an `P
-decomposition of G
such that τ (B 0 ) =
B∈B0 τ (B) <
B∈B τ (B) = τ (B),
a contradiction to the minimality of τ (B). Thus, we have
τ (B10 ) = τ (B1 ) + 1 and B̄10 contains a cycle C 0 that
contains xv . Now, consider a neighbor v 0 of x in C 0 such
that v 0 6= v , and repeat this operation while necessary
considering B10 and v 0 instead of B0 and v . We can show
that, under some conditions, after repeating this operation a
finite number of times, we obtain a better decomposition (an
`-decomposition with more trackings that induce paths than
the previous decomposition – see Lemma 3.2). In the complete version of this work, we formalize which properties
the sequence of trails must satisfy in order to guarantee this
improvement. Here, we illustrate in Figure 3 a step-by-step
of the algorithm presented above.
3.2 Hanging edges and complete tracking decompositions
All concepts defined here refer to `-decompositions B of a
graph G, for a positive integer `. Recall that any tracking
in B has exactly two end-vertices, even if they coincide.
For a tracking B in B , we denote by τ (B) the number of
end-vertices of P
B that have degree greater than 1 in B̄ ; and
define τ (B) = B∈B τ (B). Note that τ (B) ∈ {0, 1, 2}, and
τ (B) = 0 if and only if B̄ is a path. Figure 2a illustrates the
cases where τ (B) = 1, while Figures 2b–f illustrate the cases
where τ (B) = 2.
Let uv be an edge of G, and let B be the element of B
that contains uv . If B = x0 x1 · · · x` with x0 = u and x1 = v ,
or x` = u and x`−1 = v , then we say that uv is a prehanging edge at v in the decomposition B . If, moreover,
dB̄ (u) = 1, then we say that uv is a hanging edge at v in
the decomposition
B . We denote by preHang(v, B) resp.
Hang(v, B) the number of pre-hanging (resp. hanging)
edges at v in the decomposition B . Note that every hanging
edge at a vertex v is also a pre-hanging edge at v . Thus,
we have preHang(v, B) ≥ Hang(v, B) for every vertex v .
Let k be a positive integer. We say that B is k -pre-complete if
preHang(v, B) > k for every v in V (G). If Hang(v, B) > k
5
matching in a 3-regular graph G. The subgraph G−M is a 2factor of G. Choose an Eulerian orientation for G − M and,
for each edge xy in M , let Pxy be the subgraph of G that
contains xy and the edges of G − M that leave x and y with
respect to the orientation chosen. The set {Pxy : xy ∈ M } is
a P3 -decomposition of G (see Figure 4).
(a)
(b)
Fig. 4: Illustration of the proof of Kotzig’s, and Bouchet and
Fouquet’s Theorem.
(c)
(d)
Fig. 3: Illustration of how to deal with a sequence of trails
obtained by the algorithm above. At each step, the dashed
edges are the edges that are switched.
for every v in V (G), then we say that B is k -complete. We
also say that B is complete, if B is k -complete for some k .
Let J be the set of the edges of G that are initial or final
edges of trackings of B . Consider an orientation O of J with
the following property: if uv is an edge in J and B is the
element of B that contains uv , where B = x0 x1 · · · x` with
x0 = u and x1 = v , or x` = u and x`−1 = v , then uv is directed (in O) from v to u. Note that d+
O (v) = preHang(v, B)
for every vertex v of G. Moreover, for every v in G we define
−
B(v) :=
P dO (v). We note that if B is an `-decomposition of G,
then
v∈V (G) B(v) = 2|B| = 2|E(G)|/`, because every
element of B has exactly two end-vertices (counted with
multiplicity). Now, we are ready to present the Disentangling Lemma.
Lemma 3.2 (Disengangling Lemma). Let ` be a positive integer
and G be a graph. If B is a k -complete feasible `-decomposition
of G and τ (B) > 0, then there is an `-decomposition B 0 of G with
the following properties.
•
•
•
4
τ (B 0 ) < τ (B);
B 0 (v) = B(v) for every v ∈ V (G);
B 0 is k -complete.
D ECOMPOSITION OF REGULAR GRAPHS
INTO PATHS OF FIXED LENGTH
In this section we apply the Disentangling Lemma presented
in Section 3 and obtain decompositions into paths of fixed
length of a large family of regular graphs.
Kotzig [31] and, independently, Bouchet and Fouquet [14] proved that a 3-regular graph admits a P3 decomposition if and only if it contains a perfect matching.
The proof of this result introduce the structure of the proof
we use in this section, and that we generalize for highly
edge-connected graphs in section 5. Let M be a perfect
Kotzig asked what are the necessary and sufficient conditions for an odd `-regular graph G to be decomposable
in paths of length `. A necessary condition is the existence
of an b`/2c-factor (and, consequently, an d`/2e-factor) in G.
To see this, let D be a decomposition G into `-paths and,
for each path P = x0 x1 · · · x` ∈ D, consider the subset
FP = {x1 x2 , x3 x4 , . . . , x`−2 x`−1 }; the subgraph of G given
by F = ∪P ∈D FP is an b`/2c-factor of G, and the subgraph
G − E(F ) is an d`/2e-factor of G (see Figure 5).
(a)
(b)
Fig. 5: 5-regular graph decomposed (a) into copies of P5 and
(b) into a 2-factor and a 3-factor.
Favaron, Genest, and Kouider [21] showed that the
existence of an b`/2c-factor is not sufficient for G to be
decomposable into paths of length ` (see Figure 6). On the
other hand, they showed that, for a 5-regular graph to admit
a P5 -decomposition, it is sufficient for it to contain a perfect
matching and no cycles of length 4, and stated the following
conjecture.
Conjecture 4.1 (Favaron–Genest–Kouider, 2010). Let G be an
`-regular graph, ` odd. If G contains a perfect matching, then G
admits a P` -decomposition.
Recently, we proved [12] that every triangle-free 5regular graph that contains a perfect matching admits a P5 decomposition, thus verifying Conjecture 4.1 for 5-regular
graphs with girth at least 4. In Subsection ?? we present
a generalization of the result in [12]. More specifically, we
prove (see Theorem 4.13) that odd `-regular graphs with
girth at least ` − 1 and containing a perfect matching admit
P` -decompositions.
Let D be a decomposition of a graph G into trails. Given
a vertex v of G, we denote by D(v) the number of elements
of D that have v as an end-vertex. We say that D is balanced if
D(u) = D(v) for every u, v ∈ V (G). Note that D(v) ≡ d(v)
6
(i)
(ii)
Fig. 6: 5-regular graph that contains a 2-factor, but does not
admit a P5 -decomposition.
(mod 2) for every v in V (G). The following fact will be used
frequently.
Fact 4.2. Let `, m and n be positive integers.
(i)
(ii)
If ` is odd and D is a decomposition of an `-regular graph
into trails of length `, then D is balanced.
If nm is even and D is a balanced decomposition into trails
of length ` of an m`-regular graph G with n vertices, then
D(v) = m for every vertex v of G.
Heinrich, Liu and Yu [26] proved that if G is a 3mregular graph that contains an m-factor, then G admits a
balanced P3 -decompositions
in this case, D(v) = m for
every v in V (G) . Kouider and Lonc [32] proved that if G
is a 2`-regular graph with girth g ≥ (` + 3)/2, then G
admits a balanced P` -decomposition. They also proved that
if ` is even, then every bipartite `-regular graph with girth
g ≥ (` + 3)/2 admits a P` -decomposition. Moreover, they
stated the following conjecture.
Conjecture 4.3 (Kouider–Lonc, 1999). Every 2`-regular graph
admits a balanced P` -decomposition.
Thus, to solve Conjecture 4.3, it remains to prove it for
graphs with girth g < (` + 3)/2. Kouider and Lonc [32]
proved it for ` = 3. Conjecture 4.3 is a strengthening (for
paths) of Conjecture 1.2 for decompositions of 2`-regular
graphs into trees with ` edges. Recently, we proved [13] that
Conjecture 4.3 holds for ` = 4, i.e., if G is an 8-regular graph,
then G admits a balanced P4 -decomposition.
In this section, we consider the problem of obtaining
balanced P` -decompositions of m`-regular graphs, for a
positive integer m. We propose the following conjecture,
that consists of a generalization of Conjecture 4.1 (item (i))
and an equivalent form of Conjecture 4.3 (item (ii)). The
equivalence between Conjecture 4.3 and item (ii) of Conjecture 4.4 follows from Petersen’s 2-factorization Theorem
(see Theorem 2.1).
Conjecture 4.4 (Botler–Mota–Oshiro–Wakabayashi, 2015).
For m and ` positive integers, the following statements hold.
If ` is odd, then every m`-regular graph that contains an
m-factor admits a balanced P` -decomposition;
Every 2m`-regular graph admits a balanced P` decomposition.
Note that item (ii) of Conjecture 4.4 does not hold if
instead of 2m` we consider m` with m odd and ` even.
Indeed, if G is an m`-regular graph with n vertices, then
|E(G)| = nm`/2. If G admits a P` -decomposition D,
then |D| = nm/2, hence n must be even. By Fact 4.2 (ii)
D(v) = m for every v ∈ V (G), but, since D is a path
decomposition and G is Eulerian, D(v) must be even for
every v ∈ V (G), a contradiction.
In this section we prove that Conjecture 4.4 (i) holds for
graphs with girth at least g such that m > 2b(`−2)/(g −2)c,
and Conjecture 4.4 (ii) holds for graphs with girth at least g
such that m > b(` − 2)/(g − 2)c. In particular, we prove
Conjecture 4.4 (i) for m, ` such that m ≥ 2` − 1, and
Conjecture 4.4 (ii) for m, ` such that m ≥ ` − 1. Our
main results, Theorems 4.10 and 4.11, are as follows: let g , `
and m be positive integers with g ≥ 3. (i) if ` is odd
and m > 2b(` − 2)/(g − 2)c, then every m`-regular graph
with girth at least g and that contains an m-factor admits
a balanced P` -decomposition; (ii) if m > b(` − 2)/(g − 2)c,
then every 2m`-regular graph with girth at least g admits a
balanced P` -decomposition.
Note that, for ` odd, our results implies that, for every
positive integer m ≥ 3, if G is an m`-regular graph with
girth at least `−1 and containing an m-factor, then G admits
a balanced P` -decomposition. The results in [12] may be
easily generalized for proving this statement for m = 1 (see
Theorem 4.13), and a theorem in [32] deals with the case
m = 2. Furthermore, in the case G is a bipartite graph, we
prove that item (i) of Conjecture 4.4 holds for every m > `,
while item (ii) holds for every m > `/2 (see Subsection 4.3).
4.1 Completeness, feasibility and prescribed girth
In this subsection we give sufficient conditions to obtain a
k -complete `-decomposition from a (k + r)-pre-complete `decomposition (see Lemma 4.5), and to an `-decomposition
to be feasible (see Lemma 4.6).
Lemma 4.5. Let g, k, ` and r be positive integers with g ≥ 3 and
let G be a graph with girth at least g . If r ≥ b(` − 2)/(g − 2)c
and G admits a (k + r)-pre-complete `-decomposition B , then G
admits a k -complete `-decomposition B 0 such that B 0 (v) = B(v)
for every vertex v of G.
Proof. Let g, k, `, r and G be as in the hypothesis. Let B be
a (k + r)-pre-complete `-decomposition of G, and let B 0
be a (k + r)-pre-complete `-decomposition of G such that
0
B
vertex v of G, and that maximizes
P(v) = B(v) for every
0
Hang(v,
B
)
.
We
claim that B 0 is k -complete, i.e.,
v∈V (G)
0
Hang(v, B ) > k for every vertex v of G.
Suppose, for a contradiction, that B 0 is not k -complete.
Thus, there is a vertex u of G such that Hang(u, B 0 ) ≤ k .
Since B 0 is (k + r)-pre-complete, preHang(u, B 0 ) ≥ k + r + 1.
Thus, there is at least r + 1 pre-hanging edges at u that
are not hanging edges at u, say ux1 , . . . , uxr+1 . Let T1 =
y0 y1 · · · y` be the element of B 0 that contains ux1 , where,
without loss of generality, y0 = x1 and y1 = u, and let
7
X = {x1 , . . . , xr+1 }. Let x01 , . . . , x0s be the vertices in X contained in V (T1 ), ordered by distance to y1 in the sequence
y1 y2 · · · y` . Let `0 be the distance from y1 to x01 in T1 − y0 y1 ,
and let `i be the distance from x0i to x0i+1 in T1 − y0 y1 , for
0 < i ≤ s − 1 (see Figure 7). Since the girth of G is at least g ,
we have `0 ≥ g − 1, and `i ≥ g − 2 for 1 ≤ i ≤ s − 1. Since
x01 , . . . , x0s arePordered by distance to y1 in T1 − y0 y1 , we
s−1
have ` − 1 ≥ i=0 `i ≥ g − 1 + (s − 1)(g − 2) = s(g − 2) + 1.
Thus, s ≤ (` − 2)/(g − 2), which implies that s ≤ r. Since
|X| = r + 1 > s, there is at least a vertex in X , say xp , that
is not a vertex of T1 .
Let Tp be the element of B 0 that contains uxp . We can
suppose without loss of generality that Tp = z0 z1 · · · z` ,
where z0 = xp and z1 = u = y1 . Let T10 = xp y1 · · · y`
and Tp0 = x1 z1 · · · z` . Note that T̄10 = T̄1 − ux1 + uxp and
T̄p0 = T̄p −uxp +ux1 , and consider B 00 = B 0 −T1 −Tp +T10 +Tp0 .
It’s not hard no check that B 00 (v) = B 0 (v) = B(v) for
every v in V (G). Since xp ∈
/ V (T1 ), we have dT̄10 (xp ) = 1,
which implies that uxp is a hanging
edge at u in the
P
00
decomposition
B
.
Therefore,
Hang(v, B 00 ) >
v∈V
(G)
P
0
Pv∈V (G) Hang(v, B 0 ), a contradiction to the maximality of
v∈V (G) Hang(v, B ).
u
···
`0
x01
···
`1
x02
x0s−1
···
···
···
···
x0s
···
which implies that s1 + s2 = s2 ≤ (` − 2)/(g − 2). In
both cases we have s1 + s2 ≤ b(` − 2)/(g − 2)c. Since
|W | ≥ b(` − 2)/(g − 2)c + 1, we have W 6⊆ V (T ) and,
therefore, there is a vertex wi ∈ W that is not a vertex
of T .
Now, we use the lemma above to prove the main result of
this subsection, that allow us to obtain an `-decomposition
into paths from an `-decomposition b(` − 2)/(g − 2)ccomplete of graphs with girth at least g .
Lemma 4.7. Let g, k and ` be positive integers with g ≥ 3 and
let G be a graph with girth at least g . If k ≥ b(` − 2)/(g − 2)c
and B is a k -complete `-decomposition of G, then G admits a k complete `-decomposition into paths B 0 such that B(v) = B 0 (v)
for every vertex v of G.
Proof. Let g , k , `, G and B be as in the hypothesis
of the lemma. Let B be the set of all k -complete `decompositions B 0 of G such that B 0 (v) = B(v) for every
vertex v of G. From the hypothesis of the lemma, we have
B 6= ∅. Let τ ∗ = min{τ (B 0 ) : B 0 ∈ B} and let Bmin be an
element of B such that τ (Bmin ) = τ ∗ . If τ ∗ = 0, then Bmin
is an `-decomposition into paths and the proof is finished.
Thus, suppose that τ ∗ > 0. By Lemma 4.6, Bmin is a feasible a `-decomposition. Since τ (Bmin ) > 0, by Lemma 3.2
applied with k , `, G, and Bmin , there is a k -complete `decomposition B 0 of G such that τ (B 0 ) < τ (Bmin ) = τ ∗
and B 0 (v) = B(v) for every vertex v of G. Therefore, B 0 is
an element of B with τ (B 0 ) < τ (Bmin ), a contradiction to
the minimality of τ ∗ .
`s−1
Fig. 7: Obtaining a k -complete `-decomposition from a (k +
r)-pre-complete `-decomposition.
The following lemma give a sufficient condition for a
complete `-decomposition to be feasible.
Lemma 4.6. Let g, k and ` be positive integers with g ≥ 3 and
let G be a graph with girth at least g . If k ≥ b(` − 2)/(g − 2)c
and B is a k -complete `-decomposition of G, then B is feasible.
Proof. Let g , k , `, G and B be as in the hypothesis.
Fix v ∈ V (G), and let T be an `-trail of G that contains a cycle, say C , such that v ∈ V (C). Since B is
k -complete, Hang(v, B) > k ≥ b(` − 2)/(g − 2)c. Let
vw1 , . . . , vwb(`−2)/(g−2)c+1 be hanging edges at v in the
decomposition B , and let W = {w1 , . . . , wb(`−2)/(g−2)c+1 }.
We claim that there is wi ∈ W tal que wi ∈
/ V (T ). Split T
at v , obtaining two trails T1 and T2 of length c1 and c2 ,
respectively. Denote by si the number of vertices of W in Ti .
If W ⊆ V (T ), then we have |W | ≤ s1 + s2 . Suppose that
s1 ≥ 1. Let w10 , . . . , ws0 1 be the vertices of W in V (T1 ),
ordered by distance to v in T1 . Let `0 be the distance
in T1 from v to w10 , and `i be the distance in T1 from wi0
0
to wi+1
, for i > 0. Since G has girth at least g , we have
d0 ≥ P
g − 1, and `i ≥ g − 2, for 1 ≤ i ≤ s1 − 1. Thus,
1 −1
c1 ≥ si=0
`i ≥ g − 1 + (s1 − 1)(g − 2) = s1 (g − 2) + 1.
Analogously, we obtain that if s2 ≥ 1, then c2 ≥ s2 (g−2)+1.
Thus, if s1 , s2 ≥ 1, then s1 + s2 ≤ (` − 2)/(g − 2), a
contradiction. Now, suppose that s1 = 0. Since v is an
internal vertex of T , we have ` − 1 ≥ c2 ≥ s2 (g − 2) + 1,
4.2
Decompositions of regular graphs
The proof of our main results on `-decompositions follow by
induction in `. To obtain these proofs, we need first to extend
the concept of balanced decompositions to tracking decompositions. Let B be a tracking decomposition of a graph G.
Recall that we defined B(v), for v ∈ V (G), as the number of
edges of G incident to v that are initial edges of trackings in
B that start at v , or final edges of trackings in B that end at
v . Analogously to the definition of balanced decomposition
into trails, we say that B is balanced if B(u) = B(v) for every
u, v ∈ V (G). Note that B is balanced if and only if the
decomposition into trails subjacent to B is also balanced.
The two following results consist of special cases of the
theorems we prove next.
Theorem 4.8 (Heinrich–Liu–Yu [26]). Let m be a positive
integer. If G is a 3m-regular graph that contains an m-factor,
then G admits a balanced 3-decomposition into paths.
Proposition 4.9. Let m be a positive integer. If G is a 4m-regular
graph, then G admits a balanced 2-decomposition into paths.
Proof. Let m and G be as in the statement. Consider an
Eulerian orientation of G. Since G is 4m-regular, we have
d+ (v) = d− (v) = 2m for every v ∈ V (G). For each
v ∈ V (G), we decompose the set of edges that leave v into m
paths of length 2. Let B be the 2-decomposition composed
by any tracking composta of each of these 2-paths, and note
that B(v) = d− (v) = 2m for every vertex v of V (G). This
concludes the proof.
8
The following theorem is our main result about decompositions into paths of odd length.
Theorem 4.10. Let `, g and m be positive integers such that `
is odd and g ≥ 3, and let G be an m`-regular graph with girth
at least g and that contains an m-factor. If m > 2b(` − 2)/(g −
2)c, then G admits a balanced `-decomposition into paths (and,
consequently, a balanced P` -decomposition).
Proof. The proof follows by induction on `. By Theorem 4.8,
the statement holds for ` = 3 and g ≥ 3. Fix ` ≥ 5 and
suppose that the statement holds for ` − 2.
Let M be an m-factor of G. The graph G − E(M ) is
m(` − 1)-regular and, thus, by Theorem 2.1, admits a 2factorization {F1 , . . . , Fm(`−1)/2 }. Let H be the union of m
of these factors. Thus, H is a 2m-factor of G and G0 = G −
E(H) is an m(`−2)-regular graph with girth at least g . Note
that, m > 2b(` − 2)/(g − 2)c ≥ 2b(` − 4)/(g − 2)c. Therefore,
by the induction hypothesis, G0 admits a balanced (` − 2)decomposition into paths B 0 .
We claim that B 0 (v) = m for every vertex v in V (G).
Let |V (G)| = n. Note that G0 contains nm(` − 2)/2
edges
and, consequently, B 0 contains nm/2 paths. Thus,
P
0
0
0
v∈V (G) B (v) = nm. Since B is balanced, B (v) = m
for every vertex v of G (see Fact 4.2). Choose an Eulerian
0
orientation for H . Note that d+
H (v) = m = B (v). Therefore,
0
0
for each tracking B = x1 · · · x`−1 em B we can choose
edges x1 x0 and x`−1 x` of H that leave x1 and x`−1 ,
respectively, and add them to B , obtaining the `-tracking
0
B = x0 x1 · · · x`−1 x` . Since d+
H (v) = m = B (v), we can
do this operation in such a way to use each edge of H
exactly once. Let B be the `-decomposition obtained. Note
that the edges x1 x0 and x`−1 x` are pre-hanging edges, respectively, at x1 and x`−1 in the decomposition B . Moreover,
B(v) = d−
H (v) = m for every vertex v in V (G), because H
has an Eulerian orientation. Therefore, B is balanced. Let
r = b(` − 2)/(g − 2)c and k = m − r − 1. Note that
k ≥ b(` − 2)/(g − 2)c. By the definition of pre-hanging
edge, the edges of H that leave a fixed vertex v of G are
precisely the pre-hanging edges at v in the decomposition B ;
consequently, B is (m − 1)-pre-complete, i.e., (k + r)-precomplete. By Lemma 4.5 applied with g, k, `, and r, the
graph G admits a balanced k -complete `-decomposition B ∗ .
Since k ≥ b(` − 2)/(g − 2)c, by Lemma 4.7, G admits a
balanced k -complete `-decomposition into paths.
The proof of Theorem 4.11 is similar to the proof of Theorem 4.10, and uses Proposition 4.9 instead of Theorem 4.8.
Theorem 4.11. Let `, m and g ≥ 3 be positive integers. If m >
b(` − 2)/(g − 2)c, then every 2m`-regular graph with girth at
least g admits a balanced `-path tracking decomposition.
Let ` be an odd positive integer. If we consider an m`regular graph G with girth at least ` − 1, then Theorem 4.10
guarantees that G admits a decomposition into paths of
length `, for every integer m ≥ 3. The next results show
that this result also holds for m ≤ 2. The case m = 2 is a
consequence of the following result obtained by Kouider e
Lonc [32].
Theorem 4.12 (Kouider–Lonc [32]). If G is a 2`-regular graph
with girth at least g such that ` ≤ 2g − 3, then G admits a
balanced P` -decomposition.
For the case m = 1, we generalize the result given
in [12], que states that every triangle-free 5-regular graph
that contains a perfect matching admits a P5 -decomposition.
The proof is a straightforward generalization of the proof
in [12], and will be presented in a complete version of this
work.
Theorem 4.13. Let ` be an odd positive integer. If G is an `regular graph with girth at least ` − 1 and that contains a perfect
matching, then G admits an `-decomposition into paths (and,
consequently, a P` -decomposition).
4.3
Decompositions of bipartite graphs
In this subsection we show that for bipartite graphs we can
obtain better bounds for the degree of the vertices. For that,
we use versions of Lemmas 4.5, 4.6, and 4.7 proved in [10].
The following lemmas will also be used in Section 5.
Lemma 4.14. Let k , ` and r be positive integers, and G a bipartite
graph. If r ≥ b`/2c and G admits a (k + r)-pre-complete `decomposition B , then G admits a k -complete `-decomposition B 0
such that B 0 (v) = B(v) for every vertex v of G.
Proof. Let k , `, r, G = (A, B; E) and B as in the
statement of the Lemma. Let B 0 be a (k + r)-precomplete `-decomposition of G such that B 0 (v) = B(v)
for
the sum
P every vertex v0 of G, and that maximizes
0
Hang(v,
B
)
.
We
claim
that
B
is
k
-complete,
i.e.,
v∈V (G)
Hang(v, B 0 ) > k for every vertex v of G. Suppose, for a contradiction, that B 0 is not k -complete. Then, there is a vertex u
(without loss of generality) in A such that Hang(u, B 0 ) ≤ k .
Thus, there is at least r + 1 pre-hanging edges at u that
are not hanging edges at u, say ux1 , . . . , uxr+1 . Let T1 =
y0 y1 · · · , y` be the element of B 0 that contains ux1 , where,
without loss of generality, y0 = x1 and y1 = u. The vertices
of T1 in B are y0 , y2 , . . . , y2b`/2c . Since y1 y0 is not a hanging
edge, y0 ∈ {y2 , . . . , y2b`/2c }. Thus, the number of vertices
of T1 in B is at most b`/2c. Since r ≥ b`/2c, there is at least
one pre-hanging edge that is not hanging, say uxi , at u in the
decomposition B 0 such that xi ∈
/ V (T1 ). Let Ti = z0 z1 · · · z`
be the element of B 0 that contains uxi , where z0 = xi
e z1 = u. Let T10 = z0 y1 y2 · · · y` and Ti0 = y0 z1 z2 · · · z` , and
put B 00 = B 0 − T1 − Ti + T10 + Ti0 . Note that the pre-hanging
edges in B 00 are the same in B 0 , however, the edge uxi is a
hanging edge in B 00 . Thus, we conclude that B 00 (v) = B 0 (v)
for every vertex v of G. Moreover, we have Hang(u, B 00 ) ≥
Hang(u, B 0 ) + 1, and Hang(v, B 00 )P= Hang(v, B 0 ) for every
− u. Thus, v∈V (G) Hang(v, B 00 ) >
P vertex v in V (G)
0
to the maximality
v∈V (G) Hang(v,
P B ), a contradiction
0
Hang(v,
B
)
.
Therefore,
B 0 is k of the sum
v∈V (G)
complete.
Lemma 4.15. Let ` and k be positive integers and G be a bipartite
graph. If k ≥ d`/2e and B is a k -complete `-decomposition of G,
then B is feasible.
Proof. Let `, k , G and B as in the hypothesis of the
lemma. Fix v ∈ V (G), and let T be an `-trail of G
containing a cycle C such that v ∈ V (C). Since B is
k -complete, Hang(v, B) > k . Let vw1 , . . . , vwk+1 be the
hanging edges at v in the decomposition B . We claim that
there is an index 1 ≤ i ≤ k + 1 such that wi ∈
/ V (T ).
9
Let W = {w1 , . . . , wk+1 }. Let G = (A, B; E) and suppose,
without loss of generality, that v ∈ A. Since G is bipartite,
W ⊂ B . Moreover, since T contains a cycle, T contains at
most ` vertices, hence |V (T ) ∩ B| ≤ d`/2e ≤ k . But since
|W | = k + 1, we conclude that there is a vertex w ∈ W such
that w ∈
/ V (T ).
a)
b)
c)
The proof of the next lemma is analogous to the proof of
Lemma 4.7, using Lemma 4.15 instead of Lemma 4.6.
(2)
Lemma 4.16. Let k , ` be positive integers and let G be a bipartite
graph. If k ≥ d`/2e and B is a k -complete `-decomposition of G,
then G admits a k -complete `-decomposition into paths B 0 such
that B 0 (v) = B(v) for every vertex v of G.
The two following theorems are the main results of this
subsection. Again, the proof for paths of even length is
analogous to the proof for paths of odd length. Here we
show only the proof for decompositions into paths of odd
length.
Theorem 4.17. Let ` and m be positive integers such that ` is
odd, and let G be a bipartite m`-regular graph. If m > `, then G
admits a balanced `-decomposition into paths (and, consequently,
a balanced P` -decomposition).
Theorem 4.18. Let ` and m be positive integers and let G be a
bipartite 2m`-regular graph. If m > `/2, then G admits a balanced `-decomposition into paths (and, consequently, a balanced
P` -decomposition ).
Proof of Theorem 4.17. Since every bipartite regular graph
admits a decomposition into 1-factors, the statement holds
for ` = 1. Fix ` ≥ 3 and suppose that the statement holds
for `−2. Let {M1 , . . . , Mm` } be a 1-factorization of G. Let H
be a graph obtained by the union of 2m of these 1-factors,
and put G0 = G − E(H). The graph G0 is an m(` − 2)regular graph. Since m > `/2 > (` − 2)/2, by the induction
hypothesis, G0 admits a balanced (`−2)-decomposition into
paths B 0 .
By Fact 4.2, we have B 0 (v) = m for every vertex v
in V (G). Choose an Eulerian orientation to H . Note that
0
d+
H (v) = m = B (v). Thus, we can extend each tracking P
of B 0 for a vanilla tracking T by adding to P an edge leaving each of its end-vertices. Let B be the `-decomposition
obtained, and note that B(v) = d−
H (v) = m for every
vertex v in V (G). Thus, B is balanced. Let r = (` − 1)/2
and k = m − r − 1, and note that k ≥ (` + 1)/2. By the
definition of pre-hanging edge, the edges of H that leave
a fixed vertex v of G are precisely the pre-hanging edges
at v in the decomposition B ; consequently, B is (m − 1)pre-complete. By Lemma 4.14 applied with k, `, and r, the
graph G admits a balanced k -complete `-decomposition B ∗ .
Since k ≥ (` + 1)/2, by Lemma 4.16, G admits a balanced
k -complete `-decomposition into paths.
The following theorem summarizes the results presented
in this section.
Theorem 4.19. Let `, g and m be positive integers, where g ≥ 3.
Then, the following statements hold.
(1)
If ` is odd,
(3)
5
If m > 2b(`−2)/(g −2)c and G is an m`-regular
graph with girth at least g and containing an mfactor, then G admits a balanced P` -decomposition.
If g ≥ ` − 1 and G is an m`-regular graph with
girth at least g and containing an m-factor, then G
admits a balanced P` -decomposition.
If m > `, and G is a bipartite m`-regular graph,
then G admits a balanced P` -decomposition.
If m > b(` − 2)/(g − 2)c and G is a 2m`-regular
graph with girth at least g , then G admits a balanced
P` -decomposition.
If m > `/2 and G is a bipartite 2m`-regular graph, then
G admits a balanced P` -decomposition.
H IGHLY EDGE - CONNECTED GRAPHS
In this section we study the problem of decomposing highly
edge-connected graphs into paths. More specifically, we
deal with the following conjecture proposed by Barát and
Thomassen [3].
Conjecture 5.1 (Barát–Thomassen, 2006). For each tree T ,
there is a positive integer kT such that, if G is a kT -edgeconnected graph and |E(G)| is divisible by |E(T )|, then G admits
a T -decomposition.
Barát and Thomassen [3] proved that in the case where T
is the claw K1,3 , this conjecture is equivalent to Tutte’s Weak
3-flow Conjecture, posed by Jaeger [28]. They also noted that
Conjecture 5.1 does not hold when T is a graph that contains
cycle. In a series of papers, Thomassen [46], [47], [48], [49],
[50] verified Conjecture 5.1 for stars, some bistars, paths of
length 3, and paths of length 2k , for every positive integer k .
We proved Conjecture 5.1 for paths of length 5 [9] and, more
recently, for paths of every fixed length [10]. Merker proved
Conjecture 5.1 for trees with diameter 3 and 4 [36], and some
trees with diameter 5, including the path of length 5 [45].
Using a different approach, Bensmail, Harutyunyan, Le, and
Thomassé [6] also proved Conjecture 5.1 for paths of fixed
length. In this section, we present a simplification of the
proof presented in [10].
First, we consider a result proved by Barát and Gerbner [2] and, independently, by Thomassen [49], that says
that Conjecture 5.1 is equivalent to the following version for
bipartite graphs.
Conjecture 5.2. For each tree T , there is a positive integer kT0
such that, if G is a bipartite kT0 -edge-connected graph and |E(G)|
is divisible by |E(T )|, then G admits a T -decomposition.
More specifically, these authors proved the following
theorem.
Theorem 5.3 (Barát–Gerbner [2]; Thomassen [49]). Let T be a
tree with ` edges, ` > 3. The following statements are equivalent.
(i)
(ii)
There is a positive integer kT0 such that, if G is a bipartite
kT0 -edge-connected graph and |E(G)| is divisible by `,
then G admits a T -decomposition.
There is a positive integer kT such that, if G is a kT edge-connected graph and |E(G)| is divisible by `, then G
admits a T -decomposition.
Furthermore, kT ≤ 4kT0 + 16`6`+1 and if T has diameter at
most 3, then kT ≤ 4kT0 + 16`(` + 1).
10
To prove Theorem 5.3, these authors proved the following result.
Lemma 5.4. Let T be a tree with ` edges,
` > 3, and let k be a
positive integer. If G is 4k + 16`6`+1 -edge-connected, then G
can be decomposed into two graphs G1 and G2 such that
i)
ii)
G1 is bipartite and k -edge-connected;
G2 admits a T -decomposition.
Lemma 5.4 says, in other words, that if G is highly
edge-connected, then we can remove copies of T from G
in such a way to obtain a bipartite graph that still is
highly edge-connected. Thus, from now one, we suppose
that the graph G is bipartite. The main result of this
section, Corollary 5.35, says that if ` is a positive integer
and G is a bipartite 2(13` + 4r − 4)-edge-connected, where
r = max{32(` − 1), `(` + 1)}, and |E(G)| is divisible by `,
then G admits a P` -decomposition.
As well as the proofs in Section 4, the proof of the main
result of this section consists basically of a factorization step
and an induction step.
This section is organized as follows. In Subsection 5.1,
we present some results that are used in the proofs of
Subsection 5.2. In Subsection 5.2, we adapt the definitions
of factors and factorizations for highly edge-connected bipartite graphs, we define bifactorizations and prove a version of Petersen’s Theorem (Theorem 2.1) for highly edgeconnected graphs. In Subsection 5.3, we adapt the definition
of balanced decomposition for graphs that admit bifactorizations, and use Lemmas 4.14 and 4.16 to decompose these
graphs into paths of fixed length. In Subsection 5.4, we put
together the results of Subsections 5.2 and 5.3, and prove the
main result of this section.
5.1
Preliminary results
In this subsection, we define two basic operations on graphs:
vertex splitting (Subsection 5.1.1) and edge lifting (Subsection 5.1.2). Furthermore, we present classical results that relate this operations to edge-connectivity. In Subsection 5.1.3,
we show some properties of highly edge-connected graphs
that are used in the proofs of Subsection 5.2.
give us sufficient conditions for the existence of a 2k -edgeconnected detachment of a 2k -edge-connected graph.
b
a
c
f
a1
a2
b
e1
e
f
e2 e3
g
d
c
g
d
G
H
Fig. 8: A graph G and a graph H that is an {Sa , Se }detachment of G.
Lemma 5.5 (Nash–Williams [39]). Let G be a 2k -edgeconnected graph, with k ≥ 1 and V (G) = {v1 , . . . , vn }. For
each v ∈ V (G), let Sv = {dv1 , . . . , dvsv } be a subdegree sequence
for v such that dvi ≥ 2k for i = 1, . . . , sv . Then, there is a
2k -edge-connected {Sv1 , . . . , Svn }-detachment of G.
5.1.2 Edge lifting
Let G = (V, E) be a graph and u, v, w distinct vertices
of G such that uv, vw ∈ E . The multigraph G0 = V, (E \
{uv, vw})∪{uw} is called an uw-lifting (or, simply, a lifting)
at v (see Figure 9). Note that G0 may contain parallel edges
joining u and w. If for every distinct pairs x, y ∈ V \ {v}, the
maximum number of edge-disjoint paths joining x to y in G0
is the same as in G, then the lifting at v is called admissible.
If v is a vertex of degree 2, then the lifting at v is always
admissible. Such a lifting, together with the removal of v
is called a suppression of v . The next lemma is known as
Mader’s Lifting Theorem.
v
u
v
u
w
5.1.1 Vertex Splitting
Let G = (V, E) be a graph. Given a vertex v of G, a set Sv =
{d1 , . . . , dsv } with sv positive integers is called a subdegree
sequence for v if d1 +· · ·+dsv = dG (v). We say that a graph G0
is obtained from G by a (v, Sv )-splitting if G0 is composed of
G − v together with sv new vertices v1 , . . . , vsv and dG (v)
0
new
Ssv edges such that dG (vi ) = di , for 1 ≤ i ≤ sv , and
0 (vi ) = NG (v).
N
G
i=1
For a given set V 0 = {v1 , . . . , vr } with r vertices of G,
let Sv1 , . . . , Svr be subdegree sequences for v1 , . . . , vr , respectively. Let H1 , . . . , Hr be graphs obtained as follows:
H1 is obtained from G by a (v1 , Sv1 )-splitting, the graph
H2 is obtained from H1 by a (v2 , Sv2 )-splitting, and so on,
until Hr , that is obtained from Hr−1 by a (vr , Svr )-splitting.
We say that each Hi is an {Sv1 , . . . , Svi }-detachment of G.
Roughly speaking, a detachment of G is a graph obtained
by successive applications of splittings operations in vertices
of G. In Figure 8, the graph H is an {Sa , Se }-detachment
of G, where Sa = {2, 3} e Se = {2, 2, 2}. The next result
G
w
G
Fig. 9: A graph G and a graph H that is an uw-lifting of G.
Theorem 5.6 (Mader [34]). Let G be a multigraph and v a
vertex of G. If v is not a cut-vertex, dG (v) ≥ 4, and v has at least
two neighbors, then there is an admissible lifting at v .
The following lemma will be useful in the application
of Mader’s Lifting Theorem. In what follows, we denote
by pG (x, y) the maximum number of pairwise edge-disjoint
paths joining the vertices x to y in the graph G.
Lemma 5.7. Let G be a multigraph and k a positive integer. If v
is a vertex in G such that d(v) < 2k and pG (x, y) ≥ k for any
two distinct neighbors x and y of v , then v is not a cut-vertex.
Proof. Let G, k and v as in the statement. Suppose, for
a contradiction that v is a cut-vertex, and let x be y
neighbors of v in different components of G − v . Every
11
path joining x to y contains v as an internal vertex. Since
pG (x, y) ≥ k , then there are P1 , . . . , Pk pairwise edgedisjoint paths joining x to y . Since dPi (v) ≥ 2, it follows
Pk
that dG (v) ≥ i=1 dPi (v) = 2k , a contradiction.
5.1.3
High edge-connectivity
In this subsection, we present some properties shown by
graphs with given edge-connectivity.
If G is a graph that contains 2k pairwise edge-disjoint
spanning trees, then, clearly, G is 2k -edge-connected. The
converse is not true, but as stated in the next theorem, every
2k -edge-connected graph contains k such trees.
Theorem 5.8 (Nash-Williams [37]; Tutte [51]). Let k be a
positive integer. If G is a 2k -edge-connected graph, then G
contains k pairwise edge-disjoint spanning trees.
The following recent result of Lovász, Thomassen, Wu
and Zhang [33] improves a result of Thomassen [48], and allows us, under obvious necessary conditions, and a minimal
edge-connectivity, to find special orientations of the edges of
a graph.
Theorem 5.9 (Lovász–Thomassen–Wu–Zhang [33]). Let k ≥
3 a positive integer and G a (3k − 2)-edge-connected
graph.
P
Let p : V (G) → {0, . . . , k − 1} be such that v∈V (G) p(v) ≡
|E(G)| (mod k). Then, there is an orientation O of G such that
d+
O (v) ≡ p(v) (mod k), for every vertex v of G.
Combining the two results above, we prove the following lemma, that allows us to deal with highly edgeconnected bipartite graphs as regular graphs. This lemma
is a straightforward generalization of Proposition 2 in [49].
Lemma 5.10. Let k ≥ 3 and r be positive integers. If G =
(A1 , A2 ; E) is a (6k + 4r − 4)-edge-connected bipartite graph
and |E| is divisible by k , then G admits a decomposition into
two r-edge-connected spanning graphs G1 and G2 such that, the
degree in Gi of each vertex of Ai is divisible by k , for i = 1, 2.
Proof. Let k , r and G = (A1 , A2 , E) be as in the statement of
the lemma. By Theorem 5.8, G contains 3k + 2r − 2 pairwise
edge-disjoint spanning trees. Let H1 be the union of r of
these trees, let H2 be the union of other r of these trees,
and let H3 = G − E(H1 ) − E(H2 ). Clearly, H1 and H2 are
r-edge-connected, and H3 is (3k − 2)-edge-connected. Let
p : V (H3 ) → {0, . . . , k−1} be such that p(v) ≡ (k−1)dH1 (v)
(mod k) if v is a vertex of A1 , and p(v) ≡ (k − 1)dH2 (v)
(mod k) if v is a vertex of A2 . Thus, the following holds,
where the congruences are taken modulo k .
X
X
X
p(v) =
p(v) +
p(v)
v∈A1
v∈V (G)
≡
≡
≡
≡
v∈A2
(k − 1)(|E(H1 )| + |E(H2 )|)
(k − 1)(|E| − |E(H3 )|)
k (|E| − |E(H3 )|) − |E| + |E(H3 )|
|E(H3 )|.
Since H3 is a spanning (3k−2)-edge-connected subgraph
of G, by Theorem 5.9 there is an orientation O of H3 such
that d+
O (v) ≡ p(v) (mod k) for every v ∈ V (H3 ) = V (G).
For i = 1, 2 let Gi be the graph Hi together with the edges
of H3 that leave Ai with respect to the orientation O (note
= dHi (v)+d+
O (v) ≡
v in Ai , and, more-
that E = E(G1 )∪E(G2 )). Thus, dGi (v)
kdHi (v) ≡ 0 (mod k) for every vertex
over, Gi is r-edge-connected (because contains Hi ).
The following theorem is a generalization of a result of
Petersen [40] that says that every cubic graph with no cutedge contains a perfect matching.
Theorem 5.11 (Von Baebler [52] (see also [1, Theorem 2.37])).
Let r ≥ 2 be a positive integer, and G an (r − 1)-edge-connected
r-regular multigraph with an even number of vertices. Then G
contains a 1-factor.
The following results are obtained by generalizing a
technique used by Bárat and Gerbner [2], and will be used
in the proof of Lemma 5.14.
Theorem 5.12 (Theorem 20 in [19]). Let m be a positive integer.
If G is an m-edge-connected graph, then G contains a spanning
tree T such that dT (v) ≤ ddG (v)/me + 2 for every vertex v
of G.
Corollary 5.13. Let m be a positive integer. If G is an m-edgeconnected graph, then G contains a spanning tree T such that
dT (v) ≤ 4 dG (v)/m for every vertex v of G.
Proof. From the edge-connectivity of G, we have dG (v) ≥ m
for every vertex v of G. Combining this with Theorem 5.12,
we conclude that G contains a spanning tree T such that
dT (v) ≤ ddG (v)/me + 2 ≤ (dG (v)/m) + 3 ≤ 4 dG (v)/m.
Lemma 5.14. Let k , m and r be positive integers, and let G =
(A, B; E) be a bipartite graph. If G is 8md(k + r)/ke-edgeconnected and, for every v ∈ A, dG (v) is divisible by k + r,
then G admits a decomposition into two spanning graphs Gk
and Gr such that Gk is m-edge-connected and, for every vertex
r
k
dG (v) and dGr (v) = k+r
dG (v).
v ∈ A, we have dGk (v) = k+r
Proof. Let k , m, r and G = (A, B; E) be as in the hypothesis
of the lemma. Since G is 8md(k + r)/ke-edge-connected, by
Theorem 5.8 we conclude that G contains at least 4md(k +
r)/ke pairwise edge-disjoint spanning trees. Now, partition
the set of these 4md(k + r)/ke spanning trees into m set, say
T1 , . . .S, Tm , with 4d(k + r)/ke spanning trees each, and put
Gi = T ∈Ti T , for i = 1, . . . , m.
Clearly, Gi is 4d(k + r)/ke-edge-connected. By Corollary 5.13, Gi contains a spanning tree Ti such that, for every
v ∈ V (Gi ),
k
1
dGi (v) ≤
dGi (v).
dTi (v) ≤
d(k + r)/ke
k+r
0
Let G0 = ∪m
i=1 Ti . Clearly, G is m-edge-connected. Note
that, for every v ∈ V (G),
X
m
m
X
k
dG0 (v) =
dTi (v) ≤
dG (v)
k + r i=1 i
i=1
k
≤
dG (v).
k+r
0
Let Gk be the bipartite graph obtained from
G by
adding, for each vertex v in A, exactly k/(k + r) dG (v) −
0
dG0 (v) edges
of G − E(G ) that are incident to v (note that
k/(k + r) dG (v) is an integer). Therefore,
every vertex
v ∈ A has degree exactly k/(k + r) dG (v) in Gk . To
conclude the proof, put Gr = G − E(Gk ).
12
5.2
Factorizations
The main objective of this subsection is to show that some
highly edge-connected graphs admit “well-structured” decompositions, called bifactorizations, which are important
structures in the proof of the main theorem of this section
(presented in subsection 5.4).
5.2.1 Fractional factorizations
Before defining bifactorizations, we must generalize the
concepts of factors and factorizations. We extend the ideas
developed in [9] and formalize some ideas presented in [46].
Definition 5.15 (Factor). Let r and ` be positive integers and
G = (V, E) be a graph. Let X ⊂ V and F ⊂ E . We say
that F is an (X, r, `)-factor of G if dF (v) = (r/`)dG (v), for
every v ∈ X .
Note that if G is an `-regular graph, then a V (G), r, ` factor is precisely an r-factor of G.
Definition 5.16 (Fractional factorization). Let ` and k be
positive integers such that ` − k is a positive even number. Let
G = (V, E) be a graph and let X ⊂ V . We say that a partition
F = {M1 , . . . , Mk , F1 , . . . , F(`−k)/2 } of E is an (X, k, `)fractional factorization of G if the following properties hold.
•
•
Mi is an (X, 1, `)-factor of G, for 1 ≤ i ≤ k ; and
Fj is an Eulerian (X, 2, `)-factor of G, for 1 ≤ j ≤
(` − k)/2.
Note that if G contains an (X, 1, `)-factor, then dG (v) is
divisible by ` for every v ∈ X . Thus, this fact implies that,
if G admits an (X, k, `)-fractional factorization, then d(v) is
divisible by ` for every v ∈ X . The next lemma is the core
of this subsection.
Lemma 5.17. Let ` be an odd positive integer. If G = (A, B; E)
is an (` − 1)-edge-connected bipartite graph such that dG (v)
is divisible by ` for every v ∈ A, then G admits an (A, 1, `)fractional factorization.
Proof. Let ` and G = (A, B; E) be as in the hypothesis. First,
we want to apply Lemma 5.5 to obtain an (` − 1)-edgeconnected graph G0 with maximum degree 2` − 3. For that,
for each vertex v ∈ B , we choose integers sv ≥ 1 and 0 ≤
rv < `−1 such that dG (v) = (`−1)sv +rv . Put dv1 = `−1+rv
and dv2 = · · · = dvsv = ` − 1. Moreover, for every vertex
v ∈ A, we put sv = dG (v)/` and dvi = ` for 1 ≤ i ≤
sv . By Lemma 5.5 applied with parameters ` − 1 and
the
integers sv , dvi (1 ≤ i ≤ sv ) for every v ∈ V (G) , there
is an (` − 1)-edge-connected bipartite graph G0 obtained
from G by splitting of each vertex v of A into sv vertices
of degree `, and each vertex v of B into a vertex of degree
` − 1 + rv < 2` − 2 and sv − 1 vertices of degree ` − 1.
Let A0 and B 0 the set of vertices of G0 obtained from the
vertices of A and B , respectively. For ease of notation, if
v ∈ (A0 ∪ B 0 ) \ (A ∪ B) we also denote by v the vertex in
A ∪ B that originated v .
The next step is to obtain an `-regular multigraph G∗
from G0 through lifting operations. For that, we add some
edges to vertices in A0 and remove even degree vertices
from B 0 with successive applications of Mader’s Lifting
Theorem as follows. Let G00 , G01 , . . . , G0λ be a maximal sequence of graphs such that G00 = G0 and (for i ≥ 0)
G0i+1 is the graph obtained from G0i by the application of
an admissible lifting at an arbitrary vertex v ∈ B 0 with
dG0 (v) ∈
/ {1, 2, `}.
Recall that, given any two distinct vertices of G0 , say x
and y , we denote by pG0 (x, y) the maximum number of
pairwise edge-disjoint paths joining x to y in G0 . We claim
that pG0i (x, y) ≥ ` − 1 for any x, y in A0 and every i ≥ 0.
Clearly, pG00 (x, y) ≥ `−1 holds for any x, y in A0 , because G0
is (` − 1)-edge-connected. Fix i ≥ 0 and suppose that
pG0i (x, y) ≥ ` − 1 holds for any x, y in A0 . Let x, y two
vertices in A0 . Since G0i+1 is the graph obtained from G0i by
the application of an admissible lifting at a vertex v in B 0 ,
we have pG0i+1 (x, y) ≥ pG0i (x, y) ≥ ` − 1.
We claim that, if v ∈ B 0 , then dG0λ (v) ∈ {2, `}. Suppose,
for a contradiction, that there is a vertex v in B 0 such that
dG0λ (v) ∈
/ {2, `}. Note that dG0i (u) ≥ dG0i+1 (u) ≥ 2 for every
u ∈ V (G0 ) and every 0 ≤ i ≤ λ. Since dG0 (u) ≤ 2` − 3
for every u ∈ V (G0 ), we have 2 ≤ dG0i (u) ≤ 2` − 3 for
every 0 ≤ i ≤ λ. Thus, 2 ≤ dG0λ (v) ≤ 2` − 3. Since
dG0λ (v) ≤ 2` − 3, and for any two neighbors x and y
of v we have pG0λ (x, y) ≥ ` − 1, by Lemma 5.7 it follows
that v is not a cut-vertex of G0λ . Thus, by Mader’s Lifting
Theorem (Theorem 5.6) applied to G0λ , there is an admissible
lifting at v . Therefore, G00 , G01 , . . . , G0λ is not maximal, a
contradiction.
In G0λ the set B 0 may contain some vertices of degree 2.
For any such vertex v , if u and w are neighbors of v ,
we apply an uw-lifting at v , and remove the vertex v ,
i.e., we spireas v . Let G∗ be the graph obtained by the
application of this process at every vertex of degree 2
in B 0 . Note that the number of pairwise edge-disjoint paths
joining two distinct vertices of A0 does not decrease, i.e.,
pG∗ (x, y) ≥ pGλ (x, y) ≥ ` − 1 for every x, y in A0 . Clearly,
the set of vertices of G∗ that belong to B 0 is an independent
set; we denote it by B ∗ (eventually, B ∗ = ∅). Furthermore,
every vertex in B ∗ has degree `.
Claim 5.18. G∗ is (` − 1)-edge-connected.
Proof. Let Y ⊂ V (G∗ ). Suppose that there is at least one
vertex x of A0 in Y and at least one vertex y of A0 in V (G∗ )−
Y . Since there are at least ` − 1 pairwise edge-disjoint paths
joining x to y , there are at least ` − 1 edges with vertices in
both Y and V (G∗ ) − Y . Now, suppose A0 ⊂ Y (otherwise,
A0 ⊂ V (G∗ ) − Y , and we put V (G∗ ) − Y instead of Y ),
hence V (G∗ ) − Y ⊂ B ∗ . Since B ∗ is an independent set,
every edge with a vertex in V (G∗ ) − Y must have the other
vertex in A0 . Since every vertex in B ∗ has degree `, there are
at least ` edges with vertices in both Y and V (G∗ ) − Y .
We conclude that G∗ is an (` − 1)-edge-connected `regular multigraph with vertex set A0 ∪ B ∗ , where B ∗ is
an independent set.
Since every vertex of G∗ has odd degree, |V (G∗ )| is
even. By Theorem 5.11, G∗ contains a perfect matching M ∗ .
Since the multigraph J ∗ = G∗ − M ∗ is (` − 1)-regular,
Theorem 2.1 implies that J ∗ admits a 2-factorization,
∗
∗
say {F1∗ , . . . , F(`−1)/2
}. Therefore, M ∗ , F1∗ , . . . , F(`−1)/2
is a
∗
partition of E(G ).
Now, let’s go back to the bipartite graph G. Let xy be
an edge of G∗ . If x ∈ A0 e y ∈ B ∗ , then xy corresponds
to an edge of G. On the other hand, if x, y ∈ A0 , then
13
there is a vertex vxy of B 0 and two edges xvxy and vxy y
in E(G0 ). Moreover, xy was obtained by a xy -lifting at vxy
(either by the application of Mader’s Lifting Theorem or
by the repression of vertices of degree 2). Then, each edge
of G∗ represents an edge of G or a 2-path in G such that
the internal vertices of these 2-paths are always in B . For
each edge xy ∈ E(G∗ ), define f (xy) = {xy} if x ∈ A0
and y ∈ B ∗ , and f (xy) = {xvxy , vxy y} if x, y ∈ A0 . Note
that f (xy) ⊂ E(G) for each edge xy ∈ E(G∗ ). For a set
S ⊂ E(G∗ ), put f (S) = ∪e∈S f (e). The partition of E(G∗ )
∗
in M ∗ , F1∗ , . . . , F(`−1)/2
induces a partition of E(G) into
∗
M = f (M ) and Fi = f (Fi∗ ) for 1 ≤ i ≤ (` − 1)/2.
We prove that {M, F1 , . . . , F(`−1)/2 } is an (A, 1, `)fractional factorization. Fix an index i ∈ {1, . . . , (` − 1)/2}.
We show that M is an (A, 1, `)-factor of G and Fi is an
Eulerian (A, 2, `)-factor of G. Let v be a vertex of A in G
and put d0 (v) = d(v)/`. Thus, we know that v is represented
by d0 (v) vertices in G∗ . Since M ∗ is a perfect matching in G∗ ,
there are d0 (v) edges of M entering v and, since Fi∗ is a
2-factor in G∗ , there are 2d0 (v) edges of Fi incident to v .
Finally, since Fi∗ is Eulerian, the set Fi is Eulerian. This
concludes the proof.
Corollary 5.19. Let ` be an even positive integer. If G =
(A, B; E) is a 16(` − 2)-edge-connected bipartite graph such
that dG (v) is divisible by ` for every v ∈ A, then G admits
an (A, 2, `)-fractional factorization.
Proof. Let ` and G = (A, B; E) be as in the hypothesis. We
claim that G contains an (A, 1, `)-factor F such that G − F
is (` − 2)-edge-connected (note that dG−F (v) is divisible
by ` − 1 for every v ∈ A). Since G is 16(` − 2) = 8(` −
2)d`/(` − 1)e-edge-connected, by Lemma 5.14 (applied with
parameters k = ` − 1, m = ` − 2 and r = 1), the graph G
admits a decomposition into graphs Gk and Gr such that Gk
is (` − 2)-edge-connected
and dGk (v) = (` − 1)/` dG (v),
and dGr (v) = 1/` dG (v) for every v ∈ A. Thus, E(Gr ) is
an (A, 1, `)-factor. By Lemma 5.17, Gk admits an (A, 1, ` −
1)-fractional
factorization F . Thus, since dGk (v) = (` −
1)/` dG (v) for every v ∈ A, we conclude that F ∪ {E(Gr )}
is an (A, 2, `)-fractional factorization of G.
5.2.2
Bifactorizations
To obtain a decomposition of highly edge-connected bipartite graphs G into paths of fixed length `, we need
to combine fractional factorizations. More specifically, we
decompose G into graphs G1 and G2 and then we combine
a fractional factorization of G1 with a fractional factorization
of G2 . This process, called bifactorization, is defined as
follows.
Definition 5.20 (Bifactorization). Let ` and k be positive
integers such that ` − k is a positive even number, and let
G = (A1 , A2 ; E) be a bipartite graph. Let F1 , F2 families of
subsets of E and Gi = G[∪F ∈Fi F ], for i = 1, 2. We say that
F = (F1 , F2 ) is a (k, `)-bifactorization of G if the following
properties hold.
(i)
(ii)
{G1 , G2 } is a decomposition of G; and
Fi is an (Ai , k, `)-fractional factorization of Gi , for 1 ≤
i ≤ 2.
If G admits a (k, `)-bifactorization, we say that G is (k, `)bifactorable.
The next concept will be used to guarantee that G1
and G2 have minimum degrees sufficiently high.
Definition 5.21 (Strong bifactorization). Let ` be an even positive integer. Let G = (A1 , A2 ; E) be a bipartite graph
S that admits
a (2, `)-bifactorization F = (F1 , F2 ). Let Ei = F ∈Fi F for
1 ≤ i ≤ 2. We say that F is strong if dEi (v) ≥ (`/2)(`/2 + 1)
for every v in Ai for 1 ≤ i ≤ 2. If G admits a strong (2, `)bifactorization, we say that G is strongly (2, `)-bifactorable.
For ease of notation, if F belongs to F1 or F2 , then we
say that F is an element of F. In what follows, we give
sufficient conditions for a bipartite graph to be strongly
bifactorable.
Lemma 5.22. Let ` be an even positive integer. Let r =
max{16(` − 2), (`/2)(`/2 + 1)}. If G is a (6` + 4r − 4)edge-connected bipartite graph such that |E(G)| is divisible by `,
then G is matronly (2, `)-bifactorable.
Proof. Let `, r e G = (A, B; E) be as in the hypothesis. By
Lemma 5.10 (applied with ` and r), the graph G can be decomposed into two r-edge-connected spanning graphs G1
and G2 such that every vertex of A has degree divisible
by ` in G1 , and every vertex of B has degree divisible
by ` in G2 . But since r ≥ 16(` − 2), by Corollary 5.19
(applied with `), we conclude that G1 admits an (A, 2, `)fractional factorization and G2 admits a (B, 2, `)-fractional
factorization. Thus, G is (2, `)-bifactorable. Since G1 and G2
are r-edge-connected, we have dG1 (v) ≥ r ≥ (`/2)(`/2 + 1)
for every v ∈ A, and dG2 (v) ≥ r ≥ (`/2)(`/2 + 1) for every
v ∈ B , from what we conclude that G is strongly (2, `)bifactorable.
The result that will be used later is, specifically, the
following corollary.
Corollary 5.23. Let ` be a positive integer and let r =
max{32(`−1), `(`+1)}. IfG is a (12`+4r −4)-edge-connected
bipartite graph such that |E(G)| is divisible by 2`, then G is
strongly (2, 2`)-bifactorable.
Decomposition of bifactorable graphs into `-paths
5.3
In this subsection we prove that graphs that admit strong
bifactorizations can be decomposed into paths of fixed
length. Here, we do not make use of high edge-connectivity.
First, we adapt the concept of balanced decomposition of
a tricking decomposition for dealing with such graphs.
Propositions 5.25 and 5.26 deal with the base cases of our
induction step.
Definition 5.24 (Balanced tracking decompositions). Let `
be a positive integer.
Let G = (A, B; E) be a bipartite graph
that
admits
a
2,
2`
-bifactorization F = (F1 , F2 ), and let
S
Gi =
G F ∈Fi F for i = 1, 2. Let M1 ,N1 be the A, 1, 2` -factors
of F, and let M2 , N2 be the B, 1, 2` -factors of F. We say that an
`-decomposition B of G is F-balanced if the following properties
hold.
•
•
B(v) = dG1 (v)/` + dM2 (v) + dN2 (v), for every v ∈ A;
and
B(v) = dG2 (v)/` + dM1 (v) + dN1 (v), for every v ∈ B .
14
Our main goal is to prove Theorem 5.27, that guarantees that it is possible to obtain an F-balanced `complete `-decomposition into paths from a strong 2, 2` bifactorization F. First, we show that from a (2, 4)bifactorization (not necessarily strong) we can obtain a
balanced 2-bifactorization into paths.
Proposition 5.25. If G is a bipartite graph that admits a (2, 4)bifactorization F, then G admits an F-balanced 2-decomposition
into paths.
Proof. Let G = (A, B; E) and F be as in the statement.
Let F = (F1 , F2 ), where Fi = {Mi , Ni , Fi } for i = 1, 2.
Let OFi be an Eulerian orientation of G[Fi ], for i = 1, 2.
Let C1 be the set of components of G[F1 ]. Let T be an
element of C1 and BT = a0 b0 a1 b1 · · · as bs a0 a tracking
of T , where ai ∈ A, and bi ∈ B , for 1 ≤ i ≤ s. The
set BT0 = {ai bi ai+1 : 0 ≤ i ≤ s}, where as+1 = a0 , is a
2-decomposition of T in which every tracking has its endvertices in A. Thus, B10 = ∪T ∈C1 BT0 is a 2-decomposition
of G[F1 ] in which every tracking has its end-vertices in A.
Analogously, G[F2 ] admits a 2-decomposition B20 in which
every tracking has its end-vertices in B .
Let v be a vertex of A. Since M1 and N1 are (A, 1, 4)factors of G, we have dM1 (v) = dN1 (v). Thus, the number
of edges in M1 ∪N1 incident to v is even, and we can decompose the edges in M1 ∪N1 incident to v into paths of length 2
such that each path has its end-vertices in B . Choosing any
tracking of these paths, we obtain a 2-decomposition B100
of the edges in M1 ∪ N1 such that each path has its endvertices in B . Analogously, there is a 2-decomposition B200 of
the edges of M2 ∪ N2 such that each tracking has its endvertices in A.
Let B = B10 ∪ B20 ∪ B100 ∪ B200 . Note that only the trackings
in B10 and in B200 have end-vertices in A and, analogously,
only the paths in B20 and in B100 have end-vertices in B . Thus,
if v is a vertex in A, then B(v) = B10 (v)+B200 (v) = dG (v)/2+
dM2 (v) + dN2 (v), and if v is a vertex in B , then B(v) =
B20 (v) + B200 (v) = dG (v)/2 + dM1 (v) + dN1 (v). Therefore, B
is an F -balanced 2-decomposition into paths of G.
The proof of the following result is similar to part of
the proof given by Thomassen [46] for decomposition of
jiggly edge-connected graphs into paths of length 3, but here
we need to guarantee that the decomposition obtained is
balanced.
Proposition 5.26. If G is a bipartite graph that admits a (2, 6)bifactorization F, then G admits an F -balanced 3-decomposition
into paths.
Proof. Let G = (A, B; E) be a bipartite graph that admits a (2, 6)-bifactorization F = (F1 , F2 ), where Fi =
{Mi , Ni , Fi , Hi } for i = 1, 2. Let C1 be the set of components of G[F1 ∪ H1 ]. Let T be an element of C1 and
BT = a0 b0 a1 b1 · · · as bs a0 a tracking of T , where ai ∈ A and
bi ∈ B , for 1 ≤ i ≤ s. The set BT0 = {ai bi ai+1 : 0 ≤ i ≤ s},
where as+1 = a0 , is a 2-decomposition of T in which every
tracking have its end-vertices in A. Thus, B10 = ∪T ∈C1 BT0 is
a 2-decomposition of G[F1 ∪ H1 ] in which every tracking
have its end-vertices in A. Analogously, G[F2 ∪ H2 ] admits
a 2-decomposition B20 in which every tracking have its endvertices in B .
Let Gi = G[Mi ∪Ni ∪Fi ∪Hi ] for i = 1, 2. Note that, since
M1 ∪ N1 is an (A, 2, 6)-factor and F1 ∪ H1 is an (A, 4, 6)factor of G1 , we have that dF1 ∪H1 (v) = (4/6)dG1 (v) =
2dM1 ∪N1 (v), for every vertex v in A. Note also that the
number of trackings in B10 that finish at a vertex v (note
that we are not counting the trackings that start at vertices
in A) equals 21 dF1 ∪H1 (v) = dM1 ∪N1 (v). Thus, we can extend
each tracking B of B10 by adding an edge of M1 ∪ N1 at
its final vertex, obtaining a 3-decomposition into paths B1
of G1 . Analogously, we can extend each tracking B of B20 by
adding an edge of M2 ∪ N2 at its final vertex, obtaining a
3-decomposition into paths B2 of G2 .
Let B = B1 ∪ B2 . If v is a vertex of A, then the
number of trackings that have v as end-vertex is exactly
dF1 ∪H1 (v)/2+dM2 ∪N2 (v). Thus, we have B(v) = dF1 (v)/2+
dH1 (v)/2 + dM2 (v) + dN2 (v) = dG1 (v)/3 + dM2 (v) + dN2 (v).
Analogously, we have B(v) = dG2 (v)/3 + dM1 (v) + dN1 (v)
for every vertex v in B . Therefore, B is an F -balanced 3decomposition into paths of G.
Now we are ready to prove the main result of this
section.
Theorem 5.27. Let ` be a positive integer. If G is a bipartite graph
that admits a strong (2, 2`)-bifactorization F, then G admits an
F -balanced `-decomposition into paths.
Proof. The proof follows by induction on `. By Proposition 5.25, the statement holds for ` = 2; and by Proposition 5.26, the statement holds for ` = 3. Suppose
that ` ≥ 4. Let G = (A1 , A2 ; E) be a bipartite graph
that admits a strong (2, 2`)-factorizing F = (F1 , F2 ).
We claim that G admits an F -balanced `-pre-complete `decomposition. Let F1 = {M1 , N1 , F1,1 , . . . , F1,`−1
S } and
F2 = {M2 , N2 , F2,1 , . . . , F2,`−1 }, and let Gi = G F ∈Fi F
for i = 1, 2.
From now on, fix i ∈ {1, 2}. Define d∗ (v) = dGi (v)/(2`)
for every vertex v ∈ Ai . Note that dFi,j (v) = 2d∗ (v) =
2dMi (v) = 2dNi (v) for every vertex v in Ai and 1 ≤ j ≤
`−1. For j ∈ {`−2, `−1}, let OFi,j be an Eulerian orientation
forw
back
forw
of G[Fi,j ]. Let Fi,j = Fi,j
∪ Fi,j
, where Fi,j
is the set of
back
edges of Fi,j leaving Ai with respect to OFi,j , and Fi,j
is
the set of edges of Fi,j entering Ai with respect to OFi,j . Let
forw
forw
forw
G0 = G − M1 − N1 − M2 − N2 − F1,`−2
− F1,`−1
− F2,`−2
−
forw
0
back
back
F2,`−1 , and let Fi = {Fi,`−2 , Fi,`−1 , Fi,1 , . . . , Fi,`−3 }. Let
S
forw
−
G0i = G F ∈Fi0 F . Note that G0i = Gi − Mi − Ni − Fi,`−2
forw
Fi,`−1
. Then, for every v ∈ Ai , we have
dG0i (v) = dGi (v)−4d∗ (v) = 2`d∗ (v)−4d∗ (v) = 2(`−2)d∗ (v).
(1)
0
0
0
Claim 5.28. F = (F1 , F2 ) is a strong 2, 2(` − 2) bifactorization of G0 .
Proof. To prove this claim, we must prove the following.
back
back
(i) Fi,`−2
and Fi,`−1
are Ai , 1, 2(` − 2) -factors of G0i ;
(ii) Fi,j is an Eulerian Ai , 2, 2(` − 2) -factor of G0i for
j = 1, . . . , ` − 3;
(iii) dG0i (v) ≥ (` − 2)(` − 1) for every vertex v ∈ Ai .
To prove items (i) and (ii), first note that, for every
v ∈ Ai , we have dFi,`−2
= dFi,`−1
= d∗ (v) and
back (v)
back (v)
dFi,j (v) = 2d∗ (v) for every 1 ≤ j ≤ ` − 3. From (1), we
15
back
back
conclude that Fi,`−2
and Fi,`−1
are Ai , 1, 2(` − 2) -factors
0
of Gi , and Fi,j is an Ai , 2, 2(` − 2) -factor of G0i . Since F
is a (2, 2`)-bifactorization, F1,j and F2,j are Eulerian graphs
for 1 ≤ j ≤ ` − 3.
It remains to prove (iii). Since dG0i (v) = 2(` − 2)d∗ (v)
∗
e d (v) = dGi (v)/(2`) for every vertex v ∈ Ai , we have
2(`−2)
dG0i (v)
= 2` dG1 (v) for every v ∈ Ai . Since F is a strong
2, 2` -bifactorization, we have dGi (v) ≥ `(` + 1) for every
v ∈ Ai . Thus, dG0i (v) ≥ (` − 2)(` + 1) > (` − 2)(` − 1) for
every v ∈ Ai .
Since F0 is a strong 2, 2(` − 2) -bifactorization of G0 , by
the induction hypothesis, G0 admits an F0 -balanced (` − 2)decomposition into paths B 0 . Since B 0 is an F0 -balance’s (` −
2)-decomposition into paths, we have
•
•
B 0 (v) = dG01 (v)/(` − 2) + dF2,`−2
(v) + dF2,`−1
(v) for
back
back
every v ∈ A1 ;
B 0 (v) = dG02 (v)/(` − 2) + dF1,`−2
(v) + dF1,`−1
(v) for
back
back
every v ∈ A2 .
Now we want to extend each (` − 2)-tracking of B 0 to obtain
an `-decomposition of G. For that, we add edges of E(G) −
E(G0 ) at the end-vertices of the tracking of B 0 . For each
vertex v ∈ A1 (v ∈ A2 ), let Sv be the set of edges
of M1 ∪
forw
forw
forw
forw
incident to v .
M2 ∪N2 ∪F1,`−2
∪F1,`−1
N1 ∪F2,`−2
∪F2,`−1
Note that for each edge e in E(G) − E(G0 ) there
S is exactly
one vertex v ∈ V (G) such that e ∈ Sv . Then, v∈V (G) Sv =
E(G) − E(G0 ). Thus, if we prove that B 0 (v) = |Sv | for every
vertex v ∈ V (G), then we can extend each tracking B in B 0
by adding one edge at each of its end-vertices B .
Claim 5.29. B 0 (v) = |Sv |, for every v ∈ V (G).
Proof. First, note that, since F2,`−2 and F2,`−1 are Eulerian,
we have dF back (v) = dF forw (v) and dF back (v) = dF forw (v)
2,`−2
2,`−2
2,`−1
2,`−1
for every vertex v in A1 , and since F1,`−2 and F1,`−1 are
Eulerian, we have dF back (v) = dF forw (v) and dF back (v) =
1,`−2
1,`−2
1,`−1
dF1,`−1
(v) for every vertex v in A2 .
forw
For every v ∈ Ai and every 1 ≤ j ≤ ` − 3, we have
dG0i (v)/(2(` − 2)) = dFi,j (v)/2 = d∗ (v). Now, recall that
for each vertex v ∈ Ai , we have dMi (v) = dNi (v) = d∗ (v).
Thus, for each v ∈ A1 , we have
B 0 (v) = dG01 (v)/(` − 2) + dF2,`−2
(v) + dF2,`−1
(v)
back
back
= 2d∗ (v) + dF2,`−2
(v) + dF2,`−1
(v)
forw
forw
= dM1 (v) + dN1 (v) + dF2,`−2
(v) + dF2,`−1
(v)
forw
forw
= |Sv |.
Similarly, we have B 0 (v) = |Sv | for each v ∈ A2 .
We showed that every tracking B of B 0 can be extended
by adding an edge at each of its end-vertices. Let B be the
tracking decomposition obtained after this extensions. We
conclude that B is an `-decomposition of G.
Claim 5.30. B is F-balanced.
Proof. Let x0 be a vertex of A1 . First, we show that B(x0 ) ≤
dF1,`−2
(x0 ) + dF1,`−1
(x0 ) + dM2 (x0 ) + dN2 (x0 ). If there is
forw
forw
no tracking T = x0 x1 · · · x` in B , where x0 x1 is an edge of
E(G) − E(G0 ), then B(x0 ) = 0. For each such tracking T ,
by the construction of B , we know that x0 x1 is an element
of Sx1 . Since x1 is a vertex of A2 , we have Sx1 ⊂ M2 ∪ N2 ∪
forw
forw
F1,`−2
∪ F1,`−1
. Thus,
B(x0 ) ≤ dF1,`−2
(x0 ) + dF1,`−2
(x0 ) + dM2 (x0 ) + dN2 (x0 ).
forw
forw
Now, we prove that B(x0 ) ≥ dF forw (x0 ) + dF forw (x0 ) +
1,`−2
1,`−1
dM2 (x0 )+dN2 (x0 ). Note that if x0 x1 is an edge of M2 ∪N2 ∪
forw
forw
F1,`−2
∪F1,`−1
incident to x0 in A1 these are the only edges
in G that can contribute to B(x0 ) , then, by the construction
0
of B , there is a tracking Q = x1 · · · x` of a path such that the
tracking Q = x0 x1 · · · x` x0 (of a vanilla trail) belongs to B .
Thus, B(x0 ) = dF forw (x0 )+dF forw (x0 )+dM2 (x0 )+dN2 (x0 ).
1,`−2
1,`−1
Thus, for every vertex v ∈ A1 we have
forw
forw
B(v) = |F1,`−2
(v)| + |F1,`−1
(v)| + |M2 (v)| + |N2 (v)|
∗
= 2d (v) + dM2 (v) + dN2 (v)
= dG1 (v)/` + dM2 (v) + dN2 (v).
Analogously, we have B(v) = dG2 (v)/` + dM1 (v) + dN1 (v)
for each vertex v ∈ A2 . Thus, B is an F -balanced `decomposition.
Claim 5.31. B is `-pre-complete.
Proof. Let v ∈ Ai . We will show that preHang(v, B) > `.
Note that, by the construction of B , the set of pre-hanging
edges at v in the decomposition B is exactly Sv . Then,
preHang(v, B) = |Sv | = B 0 (v). Since B 0 is F-balanced,
B 0 (v) ≥ dG0i (v)/(` − 2). Thus,
preHang(v, B) = B 0 (v) ≥ dG0i (v)/(` − 2)
= 2d∗ (v) = dGi (v)/`.
Since F is a strong 2, 2` -bifactorization of G, we
have dGi (v) ≥ `(` + 1), from what we conclude that
preHang(v, B) ≥ ` + 1. Therefore, B is a `-pre-complete
`-decomposition.
Now we are able to conclude the proof. Put k = d`/2e
and r = b`/2c. Note that ` = k +r. By Lemma 4.14 with k , `,
and r, G admits a k -complete `-decomposition B 00 such that
B(v) = B 00 (v) for every vertex v of G. Since B 00 (v) = B(v)
for every vertex v of G, B 00 is F-balanced. By Lemma 4.16, G
admits a k -complete `-decomposition into paths such that
B ∗ (v) = B 00 (v) for every vertex v of G. Therefore, G admits
an F -balanced `-decomposition into paths.
5.4 Decomposition of highly edge-connected graphs
into `-paths
In this section we put together the results of Section 5.2 and
Theorem 5.27, and prove Conjecture 5.2 for paths of fixed
length. First, we prove that Conjecture 5.2 is equivalent to
the following conjecture.
Conjecture 5.32. For every tree T , there is a positive integer kT00
such that, if G is a bipartite kT00 -edge-connected graph and |E(G)|
is divisible by 2|E(T )|, then G admits a T -decomposition.
We will prove Conjecture 5.32 for the case where T
is a path. The following result shows the equivalence of
Conjecture 5.32 and Conjecture 5.2.
Theorem 5.33. Let T be a tree with ` edges, ` ≥ 3, and let k be
a positive integer. If G is a 2(k + `)-edge-connected graph such
16
that |E(G)| is divisible by |E(T )|, then there is a subgraph H of
G such that
•
•
H admits a T -decomposition; and
G0 = G − E(H) is k -edge-connected and |E(G0 )| is
divisible by 2|E(T )|.
Proof. Let G be as in the statement. If |E(G)| is divisible
by 2|E(T )|, then we put H as the empty graph Otherwise,
|E(G)| − ` is divisible by 2|E(T )|. In this case, let H be
a copy of T in G. By theorem 5.8, G contains k + ` edgedisjoint spanning trees. Since T has ` edges, H intercepts
at most ` of these trees. Thus, G − E(H) contains at least k
edge-disjoint spanning trees. Therefore, G−E(H) is k -edgeconnected.
Theorem 5.34. Let ` be a positive integer, and let r =
max{32(`−1), `(`+1)}. If G is a (12`+4r−4)-edge-connected
bipartite graph such that |E(G)| is divisible by 2`, then G admits
a P` -decomposition.
Proof. Let `, r, and G be as in the hypothesis. By Corollary 5.23, G admits a strong (2, 2`)-bifactorization F. By
Theorem 5.27, G admits an F -balanced `-decomposition
into paths. Therefore, G admits a P` -decomposition.
Corollary 5.35. Let ` a positive integer, and let r = max{32(`−
1), `(` + 1)}. If G is a 2(13` + 4r − 4)-edge-connected bipartite
graph such that |E(G)| is divisible by `, then G admits a P` decomposition.
Proof. The proof follows by applying Theorem 5.33 with k =
12` + 4r − r and Theorem 5.34.
Corollary 5.36. Let ` be a positive integer, r = max{32(` −
1), `(` + 1)}, and put kT0 = 2(13` + 4r − 4). If G is a
(4kT0 + 16`6`+1 )-edge-connected graph such that |E(G)| is
divisible by a`, then G admits a P` -decomposition.
Proof. The proof follows directly from Corollary 5.35 and
Theorem 5.3.
and Thomassé [6] obtained a similar result using a different
approach. Recently, together with Merker, these authors [5]
proved Conjecture 5.1. This shows that the study of graph
decompositions allows us to explore many different techniques that help finding partial or complete solutions to
open problems, bringing nice contributions to structural
graph theory. To illustrate this fact, we recommend [15], [23].
We plan to continue working on Conjectures 1.2 and 5.1,
and try to extend the family of trees for which they hold.
It would be very interesting to generalize the Disentangling
Lemma to deal with more general structures, because, as
observed above, this would lead to a constructive proof of
Conjecture 5.1. Furthermore, such a generalization would
allow us to obtain results analogous to the ones in Section 4
for more general classes of trees.
In another direction, we believe that it is possible to
improve the girth condition of Conjecture 4.1.
R EFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
6
C ONCLUDING REMARKS
Graph decomposition is a topic that has shown to be rich
in conjectures and challenging problems that have brought
significant contributions to structural graph theory. In this
work we developed a technique to deal with decompositions of graphs into paths that has shown to be useful to
deal with well-studied problems (Conjectures 1.2, 4.1, and
5.1). Furthermore, the tools developed in this work have
led us to other new results as in [13], obtained during the
writing of this text.
We emphasize that this work has benefited greatly from
Thomassen’s results for decompositions of highly edgeconnected graphs; we hope that this connection has become
clear to the reader familiar with this results. We also want to
mention that the result obtained by Merker [36] contributes
to the literature as an alternative to the factorization and
bifactorization results presented in this work. In particular,
if it is possible to generalize the Disentangling Lemma to
deal with more general trees, Merker’s result can be applied
to solve Conjecture 5.1 for such trees.
While we were writing the main result of Section 5
in [10], we were informed that Bensmail, Harutyunyan, Le,
[8]
[9]
[10]
[11]
[12]
[13]
[14]
J. Akiyama and M. Kano, Factors and factorizations of graphs,
ser. Lecture Notes in Mathematics. Springer, Heidelberg, 2011,
vol. 2031, proof techniques in factor theory. [Online]. Available:
http://dx.doi.org/10.1007/978-3-642-21919-1
J. Barát and D. Gerbner, “Edge-decompositions of graphs into
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