Sufficient conditions for a digraph to be supereulerian
Jørgen Bang-Jensen
∗
Alessandro Maddaloni
†
June 12, 2013
Abstract
A (di)graph is supereulerian if it contains a spanning, connected, eulerian
sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this
analogy with hamiltonian cycles and by similar results in supereulerian graph
theory, we give a number of sufficient Ore type conditions for a digraph to be
supereulerian.
Furthermore, we study the following conjecture due to Thomassé and the first
author: if the arc-connectivity of a digraph is not smaller than its independence
number, then the digraph is supereulerian. As a support for this conjecture we
prove it for digraphs that are semicomplete multipartite or quasi-transitive and
verify the analogous statement for undirected graphs.
Keywords: supereulerian digraph, spanning closed trail, degree conditions, arc-connectivity,
independence number, semicomplete multipartite digraph, quasi-transitive digraph.
1
Introduction and terminology
Notation and terminology not given below is consistent with [2].
Unless specified, we deal with simple digraphs without loops or parallel arcs, but possibly with cycles of length two. Given a digraph D, we denote by V (D), A(D) its vertex
set and arc set respectively, when D is clear from the context we simply write V and
A. We will use the letter n to indicate the number of vertices of D.
When X ⊂ V (D), we denote by DhXi the subdigraph induced by X. We write X̄
−
as an abbreviation for V (D) − X. We denote by ND+ (X), ND− (X), d+
D (X), dD (X) the
out-neighborhood, in-neighborhood, out-degree and in-degree of X in D. If D is clear
from the context, then the subscript is often omitted. When Y ⊂ V (D), we write
−
d+
Y (X) (dY (X)) to indicate the number of arcs of D with tail in X and head in Y (tail
−
in Y and head in X). The degree of X in Y , dY (X), is the sum d+
Y (X) + dY (X). Given
D1 , D2 subdigraphs of D, we denote by A(D1 , D2 ) the set of arcs of D with tail in
V (D1 ) and head in V (D2 ). For u, v ∈ V (D), the symbol µD (uv) indicates the number
∗
Department of Mathematics and Computer Science, University of Southern Denmark, Odense
DK-5230, Denmark (email: [email protected]).
†
Department of Mathematics and Computer Science, University of Southern Denmark, Odense
DK-5230, Denmark (email: [email protected])
1
of arcs from u to v, i.e. the multiplicity of the arc uv, in D (which is at most one,
unless we are dealing with a directed multigraph). Again if D is clear from the context
the subscript is omitted.
The symbols κ(D), λ(D), α(D) represent the vertex-connectivity, arc-connectivity and
independence number of D.
Given a sub(di)graph H of D, the contraction of H (into the vertex h) is the
digraph D/H with vertex set V (D) − V (H), plus a new
P vertex h and, for all u, v ∈
V (D) − V (H), µD/H (uv) = µD (uv) and µD/H (uh) = x∈V (H) µD (ux), µD/H (hv) =
P
y∈V (H) µD (yv). Given vertex disjoint subdigraphs H1 , ..., Hs of D, the contraction of
H1 , ..., Hs in D is the digraph (...(D/H1 )/...)/Hs . Clearly the resulting digraph does
not depend on the order of H1 , ..., Hs .
Let v1 , ..., vn be the vertex set of D and let H1 , ..., Hn be digraphs which are pairwise vertex disjoint. The composition D[H1 , ..., Hn ] is the digraph with vertex set
∪ni=1 V (Hi ) and arc set (∪ni=1 A(Hi )) ∪ {hi hj |hi ∈ V (Hi ), hj ∈ V (Hj ), vi vj ∈ A(D)}.
This operation also does not depend on the order of H1 , ..., Hn . We will use the term
blow up of vi into a digraph K to denote the composition D[v1 , ..., vi−1 , K, vi+1 , ..., vn ].
Given a path, or a cycle, P and two vertices a 6= b such that P contains an (a, b)path Q, we denote by P [a, b] the subpath Q, we denote by P ]a, b], P [a, b[, P, ]a, b[ the
paths Q − a, Q − b, Q − {a, b} respectively.
A walk in D is an alternating sequence W = x1 a1 x2 a2 ...xk−1 ak−1 xk of vertices xi and
arcs aj from D such that ai = xi xi+1 , for i = 1, ..., k − 1. A walk is closed if x1 = xk ,
and open otherwise. If all the arcs of a walk are distinct we call it a trail. If a trail
starts at s end ends at t, we sometimes call it (s, t)-trail.
A (di)graph is eulerian if d+ (v) = d− (v) for every vertex v, it is supereulerian if
it contains a spanning connected eulerian subgraph or, equivalently, if it contains a
spanning closed trail.
Given a digraph D, a cycle factor (eulerian factor) of D is a collection of vertexdisjoint (arc-disjoint) cycles spanning V (D).
The property of being supereulerian is at the same time a relaxation of being
eulerian and of being hamiltonian: being supereulerian means having a closed walk
covering all the vertices without using an arc twice; being eulerian means having a
closed walk covering all the arcs without using an arc twice; being hamiltonian means
having a closed walk covering all the vertices without using a vertex twice.
Supereulerian graphs have been widely studied and the literature contains several
causes and consequences (such as implications on the hamiltonicity of the line graph)
of supereulerianity. See e.g. [8], [5], [13], [17]. One of the main tools developed in
supereulerian graph theory is Catlin’s reduction theorem (Theorem 1.1):
We say that a graph H is collapsible if, for every X ⊂ V (H) of even cardinality,
H has a spanning connected subgraph HX in which the set of odd-degree vertices is
exactly X. Examples of collapsible graphs are cycles on two or three vertices.
Theorem 1.1. [8] Let H be a collapsible subgraph of a graph G. G is supereulerian if
and only if the contraction G/H is supereulerian.
2
Contrary to the case of undirected graphs, not much work has been done yet for
supereulerian digraphs: the only works we know of are [1] and [12]. The purpose of
this paper is to analyze a number of sufficient conditions for a digraph to be supereulerian. We stress that, due to the analogy with hamiltonian cycles, many of our results,
and sometimes also proof techniques, are similar to well-known results, and proofs, in
hamiltonian (di)graph theory.
The paper is organized as follows: in Section 2 we study a Chvátal Erdös type condition that has been conjectured to guarantee supereulerianity by the first author and
S. Thomassé, namely that if λ(D) ≥ α(D), then D is supereulerian. We prove that
the analogous condition is sufficient for undirected graphs and analyze it on digraphs:
in the following two subsections we prove the conjecture for the class of semicomplete
multipartite digraphs and for the class of quasi-transitive digraphs. In Section 3 we obtain sufficient degree conditions for supereulerianity similar to those for hamiltonicity
due to Woodall and Meyniel. All our results are sharp. We conclude the paper with a
small section on further remarks and open problems.
2
A Chvátal Erdös type condition
Chvátal and Erdös gave the following celebrated sufficient condition for an undirected
graph to be hamiltonian:
Theorem 2.1. [9] Let G be an undirected graph. If κ(G) ≥ α(G), then G is hamiltonian.
A weaker condition has been studied for supereulerian undirected graphs by Han,
Lai, Xiong and Yan [13]: they proved that if κ(G) ≥ α(G) − 1, then G is either
supereulerian or belongs to an infinite class of well characterized exceptional graphs.
Thomassen [20] gave an infinite family of non hamiltonian (but supereulerian) digraphs
such that κ(D) = α(D) = 2, showing that the the Chvátal-Erdös theorem does not
extend to digraphs.
We study the following conjecture, due to the first author and S. Thomassé (2011,
unpublished).
Conjecture 2.2. Let D be a digraph. If λ(D) ≥ α(D), then D is supereulerian.
It is easy to see that the above condition is not necessary, for example consider a
directed cycle on 4 vertices C4 : it is eulerian, and hence supereulerian, but λ(C4 ) = 1
and α(C4 ) = 2.
In the next sections we provide infinite families of digraphs with λ(D) = α(D) − 1 that
are not supereulerian. Hence, if true, Conjecture 2.2 would be best possible.
Let us first observe that the undirected version of Conjecture 2.2 is true.
Theorem 2.3. Let G be an undirected graph on at least three vertices. If λ(G) ≥ α(G),
then G is supereulerian.
3
Proof. First note that a complete graph on at least three vertices is clearly supereulerian
and if α(G) ≥ 4, then λ(G) ≥ 4 and again G is supereulerian, since it contains two arc
disjoint spanning trees from which an eulerian spanning subgraph can be constructed
[16]. So we are left with the case α ∈ {2, 3}.
Assume by contradiction that G is not supereulerian. We reduce G to a graph G0 of
girth at least 4 by successively contracting triangles and cycles on two vertices (recall
that these are collapsible graphs). After any contraction the independence number
does not increase and the edge-connectivity does not decrease.
By Theorem 1.1 G0 is not supereulerian. Take a connected closed trail S spanning the
maximum number of vertices of G0 . The fact that G0 has no cycle on less than four
vertices, together with α(G0 ) < 4, implies that every vertex of G0 has degree at most
3, therefore S = v1 v2 ....v|S| v1 is a cycle.
Given u ∈ V (G0 ) − V (S) 6= ∅, consider a maximum collection of edge disjoint (u, S)paths P: by the maximality of S no two such paths have the same endpoint on S and
if a path of P ends at vi , then uvi+1 ∈
/ E(G0 ). Therefore
|V (S)| > |P| ≥ λ(G0 ) ≥ α(G0 ).
Now let vi , vj ∈ V (S) be endpoints of the paths Pi , Pj ∈ P, we have vi+1 vj+1 ∈
/ E(G0 ):
otherwise S − vi vi+1 − vj vj+1 + Pi + Pj + vi+1 vj+1 would contradict the maximality of
S. But now the set {vi+1 | there is a path of P ending at vi } ∪ {u} is an independent
set on more than α(G0 ) vertices, contradiction.
Having an eulerian factor is a necessary (but in general not sufficient) condition for
being supereulerian. Using flow theory1 it is not difficult to show that the condition
λ(D) ≥ α(D) guarantees the existence of an eulerian factor.
Theorem 2.4. Let D be a digraph. If λ(D) ≥ α(D) then D has an eulerian factor.
Proof. Let D0 = ({v 0 , v 00 |v ∈ V (D)}, {v 0 v 00 |v ∈ V (D)} ∪ {w00 v 0 |wv ∈ A(D)}) be the
digraph obtained by splitting every vertex v into an ingoing part v 0 and an outgoing
part v 00 . Consider the flow network N = (D0 , l, u) with l, u being lower and upper
bounds on arcs respectively, such that
l(v 0 v 00 ) = 1, u(v 0 v 00 ) = ∞, l(u00 w0 ) = 0, u(u00 w0 ) = 1
for every v ∈ V (D), uw ∈ A(D).
There is a one to one correspondence between feasible circulations on N and eulerian
factors of D. Indeed any feasible circulation in N can be decomposed into a collection
of arc disjoint cycles (having flow 1 on every arc), using all the arcs v 0 v 00 . This is easily
translated into a collection of arc disjoint cycles spanning all the vertices in D.
Viceversa, given an eulerian factor C, we form a feasible circulation x on N , defined
by x(u00 v 0 ) = 1 if and only if uv ∈ A(C), x(v 0 v 00 ) = d+
DhCi (v) for every v.
By Hoffman’s circulation theorem [14] (see also Theorem 4.8.2 in [2]) there exists a
feasible circulation on N if (and only if)
u(S̄, S) − l(S, S̄) ≥ 0
1
For definitions see e.g. Chapter 4 in [2]
4
(1)
for every S ⊂ V (D0 ).
Consider a set S ⊂ V (D0 ) that minimizes the left hand side of (1). Observe that there
is no v ∈ V (D), such that v 00 ∈ S and v 0 ∈
/ S, because otherwise u(S̄, S) = ∞. Let
0
U := {v ∈ V (D)| v ∈ S}, W := {v ∈ V (D)| v 0 ∈ S, v 00 ∈
/ S}. We have:
u(S̄, S) ≥ d−
D (U ) + |A(W )| ≥ λ(D) + |A(W )| ≥ α(D) + |A(W )| ≥ |W | = l(S, S̄).
We get therefore the desired circulation, implying the existence of an eulerian factor
of D.
Note that Theorem 2.4 can be easily extended to directed multigraphs.
It is well known (see e.g Theorem 4.8.3 in [2]) that checking the existence of a
feasible circulation on a given network is polynomial, thus we can decide the existence
of an eulerian factor and find one, if it exists, in polynomial time. On the other hand
the problem of deciding whether a given digraph is supereulerian is NP-complete in
general [1].
We now characterize supereulerian digraphs belonging to particular classes such as
semicomplete multipartite digraphs and quasi-transitive digraphs. These characterizations imply polynomial algorithms as well as positive evidence for Conjecture 2.2.
2.1
Semicomplete multipartite digraphs
A digraph D = (V, A) is semicomplete multipartite if there is a partition V1 , V2 , . . . , Vc
of V into independent sets so that every vertex in Vi shares an arc with every vertex
in Vj for 1 ≤ i < j ≤ c. We call V1 , V2 , . . . , Vc the partite sets of D.
We will prove that for semicomplete multipartite digraphs being supereulerian is equivalent to being strong and having an eulerian factor, generalizing a result of Gutin [12]
for semicomplete bipartite digraphs and extended semicomplete digraphs (those digraphs obtained from a semicomplete graph by blowing up some of the vertices into
independent sets).
We are going to use the following easy fact.
Lemma 2.5. Let v, w1 , w2 be vertices of a semicomplete multipartite digraph D, such
that there is an arc between w1 , w2 . Then there is an arc between v and {w1 , w2 }
Proof. Suppose there is no arc between v and w1 , then they are in the same partite set
P of D. As there is an arc between w1 and w2 , the latter cannot belong to P , hence
there is an arc between v and w2 .
Lemma 2.6. Let D be a semicomplete multipartite digraph and E1 , E2 two vertex disjoint closed trails. If A(E1 , E2 ), A(E2 , E1 ) 6= ∅, then there exists a closed trail E ⊂ D
such that V (E) = V (E1 ) ∪ V (E2 ).
Proof. Consider the bipartite digraph B with partitions V (E1 ), V (E2 ) and arcs A(E1 , E2 )∪
A(E2 , E1 ). We have two possibilities:
1. Every vertex of B has positive in- and out-degree. Then, clearly, B contains a
cycle C that connects E1 and E2 , so that C ∪ E1 ∪ E2 is the desired trail.
5
2. There is a vertex of B with out-degree (or in-degree) equal to 0.
Let v1 , v2 ..., vh , v1 be a spanning trail of E1 and let w1 , w2 , ..., wk , w1 be a spanning
trail of E2 . Assume, without loss of generality, that there is a vertex of E1 with no
out-neighbor in E2 . As A(E1 , E2 ) 6= ∅, there exists another vertex of E1 with out+
neighbors in E2 . Therefore there is an index i such that d+
B (vi ) = 0 and dB (vi−1 ) >
0. Let wj ∈ NB+ (vi−1 ). By Lemma 2.5 there exists one of the arcs wj vi , wj−1 vi .
In the first case the desired trail is E := ({vi−1 wj , wj vi } ∪ E1 ∪ E2 ) − vi−1 vi , in the
second case E := ({vi−1 wj , wj−1 vi } ∪ E1 ∪ E2 ) − {vi−1 vi , wj−1 wj }.
In his PhD thesis [11] Gutin proved the following
Theorem 2.7. An extended semicomplete digraph is hamiltonian if an only if it is
strong and has a cycle factor
Using a similar approach to Gutin’s proof of Theorem 2.7 we can prove a supereulerian version of it, this version extends to the whole class of semicomplete multipartite
digraphs.
Theorem 2.8. Let D be a semicomplete multipartite digraph. D is supereulerian if
and only if it is strong and has an eulerian factor.
Proof. The first implication is immediate: if there is an eulerian spanning subgraph
D0 , then clearly it is strong and consists of the union of arc disjoint cycles spanning
V (D).
Viceversa, the following is a procedure to produce a spanning connected eulerian subdigraph of D, given an eulerian factor C.
Form a minimal collection of vertex disjoint closed trails by merging those trails of
C having common vertices. For any two closed trails E, F in the collection with
A(E, F), A(F, E) 6= ∅, join E and F into a closed trail as in Lemma 2.6.
Let E1 , ..., Eh be the collection of closed trails of D obtained after the first step is no
more applicable. Note that all the trails have at least two vertices (since an eulerian
factor only contains proper cycles). Let D0 be the digraph with vertices v1 , ..., vh and
arcs {vi vj | A(Ei , Ej ) 6= ∅} and note that D0 has no 2-cycle. By the fact that D is strong
and Lemma 2.5, D0 is a strong tournament and, thus it has an hamiltonian cycle C.
Suppose, without loss of generality, that C = v1 , v2 , ..., vh , v1 . Let w1 ∈ E1 , choose
wi ∈ Ei ∩ N + (wi−1 ) for i = 2, ..., h. Note that Ei ∩ N + (wi−1 ) 6= ∅ for all i, indeed
combining the fact that D0 has no 2-cycle and Lemma 2.5 we have that there is an arc
from wi−1 to Ei .
Now let w0 be an out neighbor of w1 in E1 . Combining again the fact that D0 has no
2-cycleSand Lemma
Sh−1 2.5 we have that one of the arcs wh w0 , wh w1 is in D. In the first
case ( i Ei ∪ i=1 wi wi+1 ∪ wh w0 ) − w1 w0 is a spanning connected eulerian subdigraph
S
S
of D. In the second case i Ei ∪ h−1
i=1 wi wi+1 ∪ wh w1 is a spanning connected eulerian
subdigraph of D
As observed previously, we can check in polynomial time whether a digraph is
strong and has an eulerian factor. Moreover, the proof of Lemma 2.6 gives a polyno6
mial procedure to construct an eulerian spanning digraph from an eulerian factor of a
semicomplete multipartite digraph. We have, thus, the following2 .
Corollary 2.9. There exists a polynomial time algorithm to decide whether a given
semicomplete multipartite digraph is supereulerian and find spanning closed trail if it
exists.
Combining Theorem 2.8 and Theorem 2.4, we verify Conjecture 2.2 for semicomplete multipartite digraphs.
Theorem 2.10. Let D be a semicomplete multipartite digraph. If λ(D) ≥ α(D), then
D is supereulerian.
We show by an example that this result is sharp. Consider the semicomplete multipartite digraph D with five partite sets U, W, W 0 , Z, Z 0 , where U has size k + 1 and
the others have size k. W has all the possible arcs from all the other partite sets and
so does W 0 . Z has all the possible arcs to all the other partite sets and so does Z 0 .
Moreover there is a matching from W to Z. Figure 1 shows an example with k = 3.
We claim that λ(D) = κ(D) = k. To see this it suffices to observe that each of the
complete bipartite subdigraphs D1 = DhZ ∪ Z 0 i and D2 = DhW ∪ W 0 i are k-strong.
This implies that D0 = DhZ ∪ Z 0 ∪ W ∪ W 0 i is also k-strong since we have a matching
of size k in one direction and all possible arcs in the other direction between W ∪ W 0
and Z ∪ Z 0 . Finally, D is obtained from D0 by adding k + 1 new vertices, each joined
to and from at least k distinct vertices from D0 , implying that D is also k-strong (see
e.g. Proposition 5.8.5 in [2]).
On the other hand D is not supereulerian since it has no eulerian factor: the vertices
in U cannot be covered by arc-disjoint cycles. Since we have κ(D) = k the example
also shows that the weaker version of Conjecture 2.2, where we replace λ(D) by κ(D)
would, if true, also be best possible in terms of the vertex-connectivity.
U
Z
W
Z0
W0
Figure 1: A non supereulerian semicomplete multipartite digraph with independence
number 3 and arc-connectivity 2. Thick arcs between sets represent complete adjacency
in the direction of the arc, double arcs indicate arcs in both directions.
2
According to [12] this was also observed by Yeo.
7
We observe that all the results of this section hold for semicomplete multipartite
directed multigraphs as well.
2.2
Quasi-transitive digraphs
A digraph D is quasi-transitive if, for every triple of distinct vertices x, y, z ∈ V (D),
with xy, yz ∈ A(D), there is at least one arc between x and z. We will use the following
result from [4] to get the so called canonical decomposition of a quasi-transitive
digraph.
Theorem 2.11. Let D be a quasi transitive digraph.
1. If D is not strong, then there exist a transitive acyclic digraph T on t vertices
and strong quasi-transitive digraphs H1 , ..., Ht such that D = T [H1 , ..., Hs ]
2. If D is strong, then there exist a strong semicomplete digraph S on s vertices and
quasi-transitive digraphs Q1 , ..., Qs such that D = S[Q1 , ..., Qs ].
One can find the above decompositions in time O(n2 ).
Given a digraph D, we define the path (trail) covering number of D, pc(D)(tc(D))
as the minimum possible number of arc-disjoint trails (vertex-disjoint paths) covering
the vertices of D. Note that trails can consist of a single vertex.
Theorem 2.12. Let D be a quasi-transitive digraph. D is supereulerian if and only
if it is strong, with canonical decomposition D = S[Q1 , ..., Qs ], and the semicomplete
directed multigraph S1 obtained from D by contracting each Qi into a single vertex vi
has an eulerian factor E 0 such that d+
DhE 0 i (vi ) ≥ tc(Qi ) for every i = 1, ..., s.
Proof. Suppose D has a spanning connected eulerian subdigraph E, then D is strong.
Let D = S[Q1 , ..., Qs ] be its canonical decomposition. The subdigraph of E contained
in Qi is the union of ti ≥ tc(Qi ) arc disjoint trails, each of these trails having in- and
out-neighbors outside of Qi , when restricted to the digraph induced by E. Hence if we
consider the spanning connected eulerian subdigraph E 0 , induced by E on S1 , we have
d+
DhE 0 i (vi ) = ti ≥ tc(Qi ) for every vi ∈ S1 .
Viceversa, assume that we have E 0 as in the statement. By the multigraph version
of Theorem 2.8, we can turn E 0 into a spanning connected eulerian subdigraph E 00
+
of S1 . From the proof of Theorem 2.8 it follows that ti := d+
DhE 00 i (vi ) ≥ dDhE 0 i (vi ).
+
00
As min(d−
D (Qi ), dD (Qi )) ≥ ti ≥ tc(Qi ), we get from E a spanning connected eulerian
subdigraph of D by replacing vi by ti arc-disjoint trails in Qi spanning V (Qi ), for every
i.
Note that it is possible to design a polynomial algorithm to calculate tc(D), for a
quasi-transitive digraph D. Therefore Theorem 2.12 implies a polynomial algorithm
for the problem of deciding whether a quasi-transitive digraph is supereulerian. (The
existence of such a polynomial algorithm was already proved by Gutin in [12]).
We now use the previous characterization to prove Conjecture 2.2 for quasi-transitive
digraphs.
8
Theorem 2.13. Let D be a quasi transitive digraph. If λ(D) ≥ α(D), then D is
supereulerian.
Proof. Let D = S[Q1 , ..., Qs ] be the canonical decomposition of D, let S1 be the semicomplete directed multigraph obtained from S by contracting each Qi into a single
vertex vi and let S2 be the directed multigraph obtained after performing vertex splitting on S1 (as in the proof of Theorem 2.4). Consider the network N = (S2 , l, u),
where l(vi0 vi00 ) = tc(Qi ), u(vi0 vi00 ) = ∞, l(vi00 vj0 ) = 0, u(vi00 vj0 ) = µS1 (vi vj ), for every
vi 6= vj ∈ V (S1 ).
Since λ(D) ≥ 1, Theorem 2.12 implies that D is supereulerian if and only if there exist
a feasible circulation on N , which, by Hoffman’s circulation theorem, means that for
every T ⊂ V (S2 ) we have
l(T, T̄ ) − u(T̄ , T ) ≤ 0
(2)
Let T be a set of minimum size among those maximizing the left hand side of (2). Define
W := {v ∈ S1 |v 0 ∈ T, v 00 ∈ T̄ }. W consists of a single vertex, because if vi 6= vj ∈ W ,
then there is an arc between them, say vi vj . We have µS1 (vi vj ) = |Qi ||Qj |. Hence
u(vi00 vj0 ) = |Qi ||Qj | ≥ tc(Qj ) = l(vj0 vj00 ).
Therefore l(T − vj0 , T̄ ∪ vj0 ) − u(T̄ ∪ vj0 , T − vj0 ) ≥ l(T, T̄ ) − u(T̄ , T ), contradicting the
minimality of the size of T . Moreover if W is empty, then l(T, T̄ ) = 0 and we have the
desired circulation. So let W = {vi }.
Using Gallai-Milgram theorem [10] (see also Theorem 13.5.2 in [2]), we have
l(T, T̄ ) = tc(Qi ) ≤ pc(Qi ) ≤ α(Qi ) ≤ α(D).
On the other hand observe that the arcs in (T̄ , T ) are of the form u00 v 0 , because the
other arcs have infinite upper capacity. Hence
u(T̄ , T ) ≥ d−
S1 (W ) ≥ λ(D).
Since α(D) ≤ λ(D) we conclude that l(T, T̄ ) ≤ u(T̄ , T ).
It follows that there exist a feasible circulation on N , which means that D is supereulerian.
To see that Theorem 2.13 is best possible consider the quasi-transitive digraph D
with vertex set given by an independent set U on k vertices, together with two complete
digraphs W, Z on k − 1 vertices and all the arcs from U to W , all the arcs from Z to
W ∪ U and a matching from W to Z. We have λ(D) = k − 1 = α(D) − 1 and D is
not supereulerian (it does not even have a eulerian factor). Figure 2 shows an example
with k = 3.
9
W
U
Z
Figure 2: A non supereulerian quasi-transitive digraph with independence number 3
and arc-connectivity 2. Thick arcs between sets represent complete adjacency in the
direction of the arc.
We observe that also the results of this section can be extended to quasi-transitive
directed multigraphs.
3
Degree conditions
The purpose of this section is to show that, as it is the case for undirected graphs, several sufficient degree conditions for hamiltonicity in digraphs can be (slightly) weakened
to become sharp sufficient conditions for supereulerianity.
A well known result in hamiltonian graph theory is Ore’s theorem.
Theorem 3.1. [19] A graph satisfying d(x) + d(y) ≥ n for every pair of non-adjacent
vertices x, y is hamiltonian.
There have been many results similar to Ore’s theorem for supereulerian graphs,
the first was due to Lesniak-Foster et al. [17], successively improved by Benhocine [5],
and finally by Catlin, who got the following (best possible) result.
Theorem 3.2. [7] A 2-edge connected graph of order at least 100 and such that d(x) +
2n
d(y) ≥
− 2 for every pair of non-adjacent vertices x, y is supereulerian.
5
On the digraph side we have the following theorems. The first one is due to Woodall.
Theorem 3.3. [21] A strong digraph D satisfying d+ (x) + d− (y) ≥ n for every ordered
pair (x, y) of non-adjacent vertices is hamiltonian.
The following are two generalizations of Woodall’s theorem, the first is due to
Meyniel and the second is due to Bang-Jensen Gutin and Li.
Theorem 3.4. [18] A strong digraph D satisfying d(x) + d(y) ≥ 2n − 1 for every pair
of non-adjacent vertices x, y is hamiltonian.
We say that an ordered pair of vertices (x, y) is dominated (dominating) if there
exists z ∈ V (D), with zx, zy ∈ A(D) (xz, yz ∈ A(D)).
10
Theorem 3.5. [3] A strong digraph such that d+ (x) + d− (y) ≥ n for every ordered
pair (x, y) of dominated or dominating non-adjacent vertices is hamiltonian.
We study when these kind of conditions are sufficient for a digraph to be supereulerian.
Lemma 3.6. Let D be a digraph. Let T be an (s, t)-trail and let T 0 be an (s0 , t0 )-path,
arc-disjoint from T . If D does not contain an (s, t)-trail with vertex set V (T ) ∪ V (T 0 ),
then
+
0
0
d−
V (T ) (s ) + dV (T ) (t ) ≤ |V (T )|.
In particular, if T 0 consists of a single vertex v, then
dV (T ) (v) ≤ |V (T )|.
+
0
0
Proof. Assume that d−
V (T ) (s ) + dV (T ) (t ) ≥ |V (T )| + 1.
If there exists v ∈ V (T ) such that vs0 ∈ A − A(T ) and t0 v ∈ A − A(T ), then
T ∪ vs0 ∪ T 0 ∪ t0 v is an (s, t)-trail with vertex set V (T ) ∪ V (T 0 ). Therefore we can
assume that the set C := {x ∈ V (T ) | xs0 ∈ A − A(T ), t0 x ∈ A − A(T )} is empty and
so V (T ) ∩ {s0 , t0 } 6= ∅. Furthermore, we assume that t0 s0 ∈
/ A − A(T ), for otherwise
T ∪ T 0 ∪ t0 s0 contains the desired (s, t)-trail.
We partition V (T ) − {s0 , t0 } into the following eight sets: U = {x ∈ V (T ) | xs0 ∈
A − A(T ), t0 x ∈
/ A}, Z = {x ∈ V (T ) | xs0 ∈
/ A, t0 x ∈ A − A(T )}, K = {x ∈
V (T ) − {s0 , t0 } | xs0 , t0 x ∈
/ A}, Ks = {x ∈ V (T ) − {t0 } | xs0 ∈ A(T ), t0 x ∈
/ A},
0
0
0
Kt = {x ∈ V (T ) − {s } | xs ∈
/ A, t x ∈ A(T )},
Ws = {x ∈ V (T ) | xs0 ∈
0
0
A(T ), t x ∈ A − A(T )},
Wt = {x ∈ V (T ) | xs ∈ A − A(T ), t0 x ∈ A(T )},
Wst = {x ∈ V (T ) | xs0 , t0 x ∈ A(T )}.
Consider a maximum cardinality collection P of arc-disjoint (s0 ∪ Wt , t0 ∪ Ws )-paths
in T . We claim that |P| ≥ |Ws | + |Wt | + |Wst | − |{s0 , t0 } ∩ V (T )| + 2µ(t0 s0 ).
To see this consider the flow network N = (V (T ), A(T ) ∪ {ts}) which we obtain
from T by adding the arc ts. Let x be the flow in N obtained by sending one unit
of flow along all arcs of A(T ) except arcs of the kind ws0 where w ∈ Ws ∪ Wst ; arcs
of the kind t0 z where z ∈ Wt ∪ Wst and except, possibly, the arc t0 s0 . Since T ∪ {ts}
is eulerian, sending flow one along every arc of N would result in a circulation (balance zero at every vertex). Thus the flow x has non-zero balance exactly for the
vertices in Ws ∪ Wt ∪ Wst ∪ {s0 , t0 } and we have3 bx (s0 ) = |Ws | + |Wst | + µ(t0 s0 ), bx (t0 ) =
−(|Wt | + |Wst | + µ(t0 s0 )), bx (w) = −1 for every w ∈ Ws and finally bx (z) = 1 for
every z inWt . We recall the well-known fact (see e.g. [2, Section 4.5]) that every flow
can be decomposed into flows along paths P1 , P2 , . . . , Pr with the property that each
path starts in a vertex with positive flow balance and ends in a vertex with negative
flow balance. At most one of these paths can use the arc ts, therefore the number of
arc-disjoint (s0 ∪ Wt , t0 ∪ Ws )-paths in T is at least |Ws | + |Wt | + |Wst | + µ(t0 s0 ) − 1.
Recall that µ(t0 s0 ) = 1 if t0 s0 is an arc (of T ) and 0 otherwise and recall that the balance bx (v) of
a vertex v with respect to a flow x is the sum of the x-values on arcs leaving v minus the x-values on
the arcs entering v.
3
11
Now the claim follows by observing that if t0 s0 ∈ A, then t0 s0 ∈ A(T ), so µ(t0 s0 ) − 1 ≥
2µ(t0 s0 ) − |{s0 , t0 } ∩ V (T )|.
By our hypothesis on the degrees of s0 , t0 and the fact that C = ∅, we have
=
2(|Ws |
−
dV (T ) (s0 )
+ |Wt | + |Wst |) + |U | + |Z| + |Ks | + |Kt | =
0
0 0
0 0
+ d+
V (T ) (t ) − 2µ(t s ) > |V (T )| − 2µ(t s ) =
= |Ws | + |Wt | + |Wst | + |U | + |Z| + |Ks | + |Kt | + |K| + |{s0 , t0 } ∩ V (T )| − 2µ(t0 s0 ),
implying
|K| < |Ws | + |Wt | + |Wst | − |{s0 , t0 } ∩ V (T )| + 2µ(t0 s0 ) ≤ |P|.
(3)
If there is P ∈ P with V (P ) ∩ K = ∅, then we find an (s, t)-trail with vertex set
V (T ) ∪ V (T 0 ). Indeed let u ∈ {s0 } ∪ Wt ∪ U , z ∈ {t0 } ∪ Ws ∪ Z such that P ]u, z[ does
not contain vertices of K ∪ U ∪ Z ∪ {s0 , t0 }. Such u, z exist given that V (P ) ∩ K is
empty and given that P starts in {s0 } ∪ Wt ∪ U and ends in {t0 } ∪ Ws ∪ Z. Now all the
vertices in P ]u, z[ are adjacent with {s0 , t0 } in T , and T ∪ us0 ∪ T 0 ∪ t0 z − A(P [u, z]) is
an (s, t)-trail4 with vertex set V (T ) ∪ V (T 0 ).
Therefore we can assume that every path of P contains a vertex of K. Consider the
collection of pairwise disjoint sets (KP )P ∈P defined as KP = {v ∈ K∩P | dDhT i (v) = 2}:
we have
X
|KP | ≤ |K| < |P|,
P ∈P
where the last inequality is due to (3). We infer the existence of a path R ∈ P such
that
dV (T ) (v) > 2 ∀v ∈ R ∩ K.
(4)
We can now proceed as before to construct our (s, t)-trail: we find u ∈ {s0 } ∪ Wt ∪ U ,
z ∈ {t0 } ∪ Ws ∪ Z such that R]u, z[ does not contain vertices of U ∪ Z ∪ {s0 , t0 }. By
(4) we have that the vertices of R]u, z[ have degree more than two in T , therefore
T ∪ us0 ∪ T 0 ∪ t0 z − A(R[u, z]) is an (s, t)-trail with vertex set V (T ) ∪ V (T 0 ).
The following is an analogue of Meyniel’s theorem. Our proof uses an approach
similar to the one used in [6].
Theorem 3.7. A strong digraph such that d(x) + d(y) ≥ 2n − 3 for every pair of
non-adjacent vertices x, y is supereulerian.
Proof. Let S be a closed trail maximizing the number of vertices spanned, over all the
closed trails and let s = |V (S)|. Suppose by contradiction that s < n.
Since D is strong, there exists an (S, S)-path Q on at least three vertices. Let Q be
chosen so that the length of the shortest path P in S between the endpoints of Q is
minimum. Let x0 , v, y be the first, second and last vertex of Q. Note that, by the
maximality of S, y cannot equal x0 nor a vertex adjacent from x0 in S (in the latter
4
The case u = s0 or z = t0 is covered by considering the arc us0 or t0 z as empty.
12
case we could replace x0 y by Q).
Let W = {x1 , ..., xp } be the (non-empty) set of internal vertices of P . By our choice of
Q we have
dW (v) = 0.
(5)
Moreover, for every 1 ≤ i ≤ p, there is no vertex u ∈ V − V (S) such that xi u, uv ∈
A(D), or vu, uxi ∈ A(D). Thus
dV −V (S) (v) + dV −V (S) (xi ) ≤ 2(n − s − 1),
(6)
for every 1 ≤ i ≤ p.
Let S 0 be the vertex set of the (y, x0 )-trail obtained from S by removing the arcs of P
(and keeping only vertices left with positive degree). The cardinality of S 0 is s − p + c,
where c := |W ∩ S 0 |. By the maximality of S, Equation (5) and Lemma 3.6 (applied
to S 0 , {v}) we have
dV (S) (v) ≤ dW (v) + dS 0 (v) ≤ s − p + c.
(7)
Let b be the maximum integer 1 ≤ i ≤ p such that D contains a (y, x0 )-trail T with
vertex set S 00 := S 0 ∪ {x0 , ..., xb−1 }. Again by the maximality of S, we have p > b ≥ 1.
By the maximality of b and Lemma 3.6 (applied to T, {xb }) we have (note that xb ∈
/ S 00 )
dV (S) (xb ) = dW −S 00 (xb )+dS 00 (xb ) ≤ 2(p−1−|S 00 ∩W |)+s−p+|S 00 ∩W | ≤ s+p−c−2. (8)
Combining (6)-(8) we get:
d(v) + d(xb ) ≤ (s − p + c) + (s + p − c − 2) + 2(n − s − 1) ≤ 2n − 4,
but, by (5) v and xb are not adjacent, contradiction.
Theorem 3.7 implies the following Woodall-type sufficient condition.
Corollary 3.8. A strong digraph such that d+ (x) + d− (y) ≥ n − 1 for all ordered pairs
(x, y) of non-adjacent vertices is supereulerian.
We strengthen this condition obtaining an analogue of Theorem 3.5.
Theorem 3.9. A strong digraph such that d+ (x) + d− (y) ≥ n − 1 for every ordered
pair (x, y) of dominated or dominating non-adjacent vertices is supereulerian.
Proof. Let S be a closed trail maximizing the number of vertices spanned, over all the
closed trails and let s = |V (S)|. Suppose by contradiction that s < n.
Since D is strong, there exist x ∈ V (S), v ∈ V − V (S), such that xv ∈ A(D). We show
that there exists x0 ∈ V (S) such that vx0 ∈ A(D). Observe that v cannot be dominated
by S, otherwise the existence of a (v, S)-path contradicts the maximality of S. Thus
either there exists an x0 as above and we are done, or there exist y, y 0 ∈ V (S) with yy 0 ∈
+
0
A(S), yv ∈ A(D) and v, y 0 non-adjacent. We have that d−
V −V (S) (y ) + dV −V (S) (v) ≤
n − s − 1, because if c ∈ N − (y 0 ) ∩ N + (v) ∩ (V − V (S)), then S ∪ {yv, vc, cy 0 } − yy 0 is
a closed trail, contradicting the maximality of S. Therefore, by the hypothesis of the
+
0
theorem, d−
V (S) (y ) + dV (S) (v) ≥ s, thus v has an out-neighbor on S.
Let x, x0 ∈ V (S), v ∈ V − V (S) be chosen so that xv, vx0 ∈ A(D) and so that the
13
length of the shortest (x, x0 )-path P in S is minimum. Let W = {x1 , ..., xp } be the set
of vertices of P ]x, x0 [. By the minimality of P
dW (v) = 0.
(9)
Let S 0 be the set of vertices of the trail obtained by removing from S the arcs of P ]x, x0 [)
(and keeping only vertices left with positive degree). We have |S 0 | = s − p + c, where
c = |S 0 ∩ W | < p. By the maximality of S, Equation (9) and Lemma 3.6 applied to
S 0 , P and to S 0 , v we have
−
d+
V (S) (xp ) + dV (S) (x1 ) ≤ s − p + c + 2(p − 1 − c),
(10)
dV (S) (v) ≤ s − p + c.
(11)
Moreover, by the maximality of S, there is no vertex y ∈ V − V (S) such that xp y, yv ∈
A(D) or vy, yx1 ∈ A(D). Therefore
−
dV −V (S) (v) + d+
V −V (S) (xp ) + dV −V (S) (x1 ) ≤ 2(n − s − 1).
(12)
Putting together (10)-(12) we have
d(v) + d+ (xp ) + d− (x1 ) ≤ 2(p − 1 − c) + 2(s − p + c) + 2(n − s − 1) ≤ 2n − 4,
thus d+ (xp ) + d− (v) ≤ n − 2 or d+ (v) + d− (x1 ) ≤ n − 2, but the pair v, x1 is dominated
and the pair xp , v is dominating, contradicting the hypothesis.
As we show with an example the condition d(x) + d(y) ≥ 2n − 4 for all non-adjacent
pairs x, y does not necessarily imply being supereulerian and neither does the condition
d+ (x) + d− (y) ≥ n − 2 for all non-adjacent ordered pairs (x, y). Theorems 3.7 and 3.9
are thus best possible.
The infinite class of digraphs of Figure 3 is not supereulerian. Indeed the only inneighbor of {x, y} is z 0 , therefore any spanning subdigraph contains the arcs z 0 x, z 0 y,
but d− (z 0 ) = 1, thus such a spanning subdigraph cannot be eulerian. Moreover x, y
is the only pair of non-adjacent vertices and it is easy to see that d+ (x) + d− (y) =
d− (x) + d+ (y) = n − 2.
Note that the digraph of Figure 3 does not even have an eulerian factor. This necessary
condition however does not help to get a better bound on the degree sum of the previous
theorems, indeed from the digraph of Figure 3 we can get a counterexample with an
eulerian factor and the same degree properties, by blowing up x into a 2-cycle.
14
x
z0
y
z
Kn−4
Figure 3: A non-supereulerian digraph with d+ (x) + d− (y) ≥ n − 2 for every ordered
pair of non-adjacent vertices x, y.
4
Future work
In [15] Jackson and Ordaz proposed the following
Problem 4.1. Does there exist a function f (α), such that every digraph D with κ(D) ≥
f (α(D)) is hamiltonian?
To the best of our knowledge this problem is still open. We propose a weaker version
of Conjecture 2.2, that would follow by a positive answer of the above problem.
Conjecture 4.2. There exists a function g(α), such that every digraph D with λ(D) ≥
g(α(D)) is supereulerian.
Catlin’s reduction theorem (Theorem 1.1) has proved extremely useful in supereulerian graph theory. Unfortunately it does not seem easy to build a similar tool for
digraphs, since to infer supereulerianity on digraphs what matters are the exact degrees and not only their parity. It would be interesting, though, to develop a more
solid theory to study supereulerian digraphs.
In this paper we have focused on some causes of supereulerianity, but it could be
worth to investigate meaningful consequences of it.
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