On Variations of the 1-2-3 Conjecture
Emma Barme
École Normale Supérieure de Lyon
MC2 Team – Laboratoire d’Informatique du Parallèlisme
Supervised by Julien Bensmail
August 27, 2015
1
Introduction
This report presents the work from my Bachelor internship in the MC2
(Modèles de calcul, Complexité, Combinatoire) team at the LIP (Laboratoire d’Informatique du Parallèlisme) at the École Normale Supérieure de
Lyon. During this 6-week internship, I studied variations of the 1-2-3 Conjecture introduced by Karoński, Łuczak and Thomason in [6].
The 1-2-3 Conjecture postulates that if G is an undirected simple graph
with no component isomorphic to K2 , then there exists an assignation of
weights from the set {1, 2, 3} to the edges of G, so that for any edge uv of G,
the sum of the weights of the edges incident to u is different from the sum of
the weights of the edges incident to v. In other words, we want the incident
sums to yield a proper vertex coloring.
This conjecture stems from the search for a definition of irregularity in graphs.
As regular graphs are graphs in which all vertices have the same degree, a
possible opposite definition for irregular graphs could be graphs in which no
two vertices have the same degree. However, it is relatively easy to see that
every simple graph with at least two vertices cannot be irregular according to
this definition: if a simple graph G of order n is totally irregular, its vertices
will have degrees 0, 1, ... n − 1, but it is impossible for G to have both a
0-degree vertex and a (n − 1)-degree vertex.
As multigraphs do not present the same issue and can be irregular, one can
try to approximate the irregularity strength of a simple graph G by multiplying its edges and building an irregular multigraph with the same structure
as G. This is equivalent to weighting the edges of G and assigning to every
vertex of G a weighted degree, which is the sum of the weights of its incident
edges. This way, it possible to obtain a simple graph in which no two vertices
have the same weighted degree, as shown in Figure 1
However, since this definition of irregularity is too strong for simple graphs,
another antonym for regularity can be considered. A regular graph can also
be viewed as a graph where every two adjacent vertices have the same degree.
From this, one can define local irregularity: a graph G is locally irregular if
every two adjacent vertices of G have different colors. This leads to a relaxed
problem in which the weighted graph must not have two adjacent vertices
with the same weighted degree. Now, the main question is, given a graph G,
to guess the minimum number of weights k necessary to weight edges with
weights in {1, ..., k}, so that the weighted degrees make up a locally irregular
1
Figure 1: From an unweighted simple graph to an irregular multigraph to an
irregular weighted simple graph.
coloring? We denote the value of this minimum k by χeP , where χ indicates
P
a chromatic parameter, the subscript
states the color function is a sum
function and the superscript e states the weighted elements are the edges of
the graph.
According to the 1-2-3 Conjecture, we would have χeP (G) ≤ 3 for any
graph G with no component isomorphic to K2 (no component with only two
vertices and one edge). As of now, it has been proved that χeP (G) ≤ 5 holds
for every such graph G [5].
A number of variations of this relaxed problem are worth considering.
For example, the edge-weighting can be extended to a total-weighting, that
is a weighting of both the vertices and the edges; the weighted degree can be
replaced by the product of the weights on the incident edges; the graphs can
be oriented or directed... An overview of variations present in the literature
can be found for example in Bensmail’s PhD thesis [2] or in a survey by
Seamone [9].
During my first two weeks, I worked on the multiplicative total-weighting
variation, which had already been introduced, and I confirmed the multiplicative conjecture for a new class of graphs, namely 4-regular graphs. I then
focused on a new directed version of the 1-2-3 Conjecture, of which I have
notably proved a weaker version. In the next pages, you will find an introduction of the necessary definitions, followed by a presentation of the state
of art and of my contributions, first for the multiplicative total-weighting
variation and then for the directed variation.
2
2
Definitions and notations
In this section, we will introduce the necessary definitions and notations
to study the chosen variations. More definitions regarding basic notions in
graph theory can be found in Appendix B.
2.1
Weightings
First, we define several notations for graph weighting. In the considered
problems, graphs can have weights assigned either to just their edges (edgeweighting), or to their edges and their vertices (total-weighting). The weights
assigned are positive integers, and the set from which they are taken is defined
either by its greatest element k (and is then {1, ...k}) or explicitly.
Definition 1 (Le -edge-weighting). An Le -edge-weighting of a simple graph
G is a function w that assigns to each edge uv of G an integer weight w(uv)
in the set Le .
Definition 2 (k-edge-weighting). An k-edge-weighting is a {1, ..., k}-edgeweighting.
Definition 3 ((Lv , Le )-total-weighting). An (Lv , Le )-total-weighting of a
simple graph G is a function w that assigns to each vertex u of G an integer
weight w(u) in the set Lv and each edge uv of G an integer weight w(uv) in
the set Le .
Definition 4 (k-total-weighting). An k-total-weighting is a ({1, ..., k},
{1, ..., k})-total-weighting.
Definition 5 (La -arc-weighting). An La -arc-weighting of a simple digraph
→ of D an integer weight w(−
→
D is a function w that assigns to each arc −
uv
uv)
in the set La .
Definition 6 (k-arc-weighting). An k-arc-weighting is a {1, ..., k}-arcweighting.
2.2
Colorings
For problems surrounding the 1-2-3 Conjecture, the colors of the vertices
are defined as functions of the weights of the graph. The following sections
focus on variations with the sum function and the product function. Several
notations are also introduced, following the same pattern as χeP .
3
2.2.1
Undirected graphs
Definition 7 (Sum-coloring k-edge-weighting). Let w be a k-edge-weighting
of some undirected graph G. The sum color of a vertex u of G is
X
s(u) =
w(uu)
uv∈G
We say w is sum-coloring if for all edges uv of G, we have s(u) 6= s(v). If a
sum-coloring k-edge-weighting exists for G, then G is sum-coloring k-edgeweightable. The smallest k such that G is sum-coloring k-edge-weightable is
denoted χeP (G).
Definition 8 (Sum-coloring k-total-weighting). Let w be a k-totalweighting of some undirected graph G. The sum color of a vertex u of
G is
X
w(uv)
s(u) = w(u) +
uv∈G
We say w is sum-coloring if for all edges uv of G, we have s(u) 6= s(v). If a
sum-coloring k-total-weighting exists for G, then G is sum-coloring k-totalweightable. The smallest k such that G is sum-coloring k-total-weightable is
denoted χtP (G).
Definition 9 (Product-coloring k-edge-weighting). Let w be a k-edgeweighting of some undirected graph G. The product color of a vertex u of G
is
Y
p(u) =
w(uv)
uv∈G
We say w is product-coloring if for all edges uv of G, we have p(u) 6= p(v). If
a product-coloring k-edge-weighting exists for G, then G is product-coloring
k-edge-weightable. The smallest k such that G is product-coloring k-edgeweightable is denoted χeQ (G).
Definition 10 (Product-coloring k-total-weighting). Let w be a k-totalweighting of some undirected graph G. The product color of a vertex u of G
is
Y
p(u) = w(u) +
w(uv)
uv∈G
We say w is product-coloring if for all edges uv of G, we have p(u) 6= p(v). If
a product-coloring k-total-weighting exists for G, then G is product-coloring
4
k-total-weightable. The smallest k such that G is product-coloring k-totalweightable is denoted χtQ (G).
2.2.2
Digraphs
For digraphs, we define two colors per vertex, in order to differentiate the
color stemming from incoming arcs and the color stemming from outgoing
arcs.
Definition 11 (Sum-coloring k-arc-weighting). The sum-color of a vertex u of D, where D is a simple digraph with a k-arc-weighting w, is a couple
(s− (u), s+ (u)) of an in-color and an out-color, with
X
→
w(−
vu)
s− (u) =
−
→
vu∈G
and
s+ (u) =
X
→
w(−
uv)
−
→
uv∈G
→ ∈ D, we have s+ (u) 6= s− (u). If
We say w is sum-coloring if for all −
uv
a sum-coloring k-arc-weighting exists for G, then G is sum-coloring k-arcweightable. The smallest k such that D is sum-coloring k-arc-weightable is
denoted χaP (G).
±
An equivalent definition can of course be formulated with the product
function.
2.3
Forbidden patterns
There are some graph patterns that are never sum-coloring or multiplicativecoloring k-edge/arc-weightable. For example, complete graphs of size 2 are
not sum-coloring or multiplicative-coloring k-edge-weightable.
Definition 12 (Nice graph). A nice graph is a simple graph with no component isomorphic to K2 .
Definition 13 (Nice digraph). A nice digraph D is a simple digraph in
→ ∈ D, we have d+ (u) > 1 or d− (v) > 1.
which for every arc −
uv
5
One can easily convince oneself that graphs (resp. digraphs) with no K2
→ such that d+ (u) = d− (v) = 1) are always
components (resp. with no arc −
uv
sum-coloring or product-coloring k-edge-weightable (resp. k-arc-weightable)
if k is big enough.
For k-total-weighting, there are no forbidden patterns, all graphs are sumcoloring or product-coloring k-total-weightable if k is big enough.
3
3.1
Product-coloring k-total-weighting
Multiplicative 1-2-3 Conjecture
Skowronek-Kaziów have introduced an equivalent of the 1-2-3 Conjecture for
product-coloring k-total-weighting [10, 11]:
Conjecture 1 (Multiplicative 1-2-3 Conjecture). For every nice graph
G, we have χeQ (G) ≤ 3.
While for the sum version of the problem, it has been shown that every
nice graph is sum-coloring 5-edge-weightable, Skowronek-Kaziów has proved
that every nice graph is product-coloring 4-edge-weightable [11]. This difference rests upon some interesting properties of the multiplicative variation of
the 1-2-3 Conjecture, such as the presence of the neutral element 1 in the set
from which weights are taken, or the fact that the set of weights incident to
a vertex matches the prime factorization of the vertex color.
In addition to this, the Multiplicative 1-2-3 Conjecture holds for nice cycles,
paths, complete graphs and 3-colorable graphs [11].
3.2
Multiplicative 1-2 Conjecture
A similar conjecture exists for product-coloring total-weighting [10], as a
variation of the one introduced for sum-coloring total-weighting [8]. Because
weighting the vertices adds a degree of freedom and enables us to modify the
color of a vertex without modifying those of its neighbors, we expect χtQ to
be lower than χeQ :
Conjecture 2 (Multiplicative 1-2 Conjecture). For every nice graph G,
we have χtQ (G) ≤ 2.
6
The Multiplicative 1-2 Conjecture holds for complete graphs and 3-colorable
graphs, and every nice graph is product-coloring 3-total-weightable [10].
3.3
Contribution
Here, we show that the Multiplicative 1-2 Conjecture also holds for another
class of graphs. Our main motivation was to prove that the conjecture holds
for 4-colorable graphs, which turned out to be a difficult challenge. However,
we were able to adapt the proof by Przybyło and Woźniak that χtP (G) ≤ 2
for every 4-regular graph G [8].
Theorem 3. For every 4-regular graph G, we have χtQ (G) ≤ 2.
Proof. Let G be a 4-regular graph. If G is complete, or G is 3-colorable,
we know G is product-coloring 2-total-weightable. If G is not connected,
each connected component can be treated independently. Therefore, we can
assume G is connected and not 3-colorable without loss of generality. By
Brooks’ Theorem, G is 4-colorable.
Amongst possible colorings (A, B, C, D) of G, where A, B, C, D are the color
classes by decreasing size, we choose a coloring that maximizes the size of A,
the size of B when A is fixed, and the size of C when A and B are fixed.
By construction, each vertex in D has at least one neighbor in A, in B and
in C, each vertex in C has at least one neighbor in A and in B, and each
vertex in B has at least one neighbor in A. Let D1 be the subset of vertices
of D that have exactly one neighbor in A, and D2 the subset of vertices of D
that have exactly two neighbors in A, with D1 ∪ D2 = D. We want to build
a total-weighting w compliant with the following criteria, as illustrated by
Figure 2:
• For all u ∈ A, we have p(u) = 32;
• For all u ∈ D1 , we have p(u) = 2;
• For all u ∈ D2 , we have p(u) = 8;
• For all u ∈ B, we have p(u) = 4 or p(u) = 16;
• For all u ∈ C, we have p(u) = 1, or p(u) = 4 or p(u) = 16;
• For all uv ∈ G, we have p(u) 6= p(v).
7
Figure 2: Final weights of the color classes in the product-coloring totalweighting of 4-regular simple graphs.
We obtain this by applying the following rules:
• For all u ∈ A and for all v ∈ B ∪C ∪D, we set w(u) = 2 and w(uv) = 2;
• For all u ∈ D and for all v ∈ B ∪ C, we set w(uv) = 1;
• For all u ∈ Di , with i ∈ 1, 2, we set w(u) = i;
• For each edge (u, v) with u ∈ B and v ∈ C, we set w(uv) = 2 if v has
at least one neighbor in D1 , and w(uv) = 1 otherwise;
• For each u ∈ B, we choose w(u) so that p(u) = 2k with k even;
• For each u ∈ C, if w assigns 1 to three of the four edges incident to u,
we set w(u) = 1. We then have p(u) = 2, which works because u is not
a neighbor of any vertex v ∈ D1 ;
• For each u ∈ C, if w assigns 2 to at least two of the four edges incident
to u and u is not a neighbor of any vertex v ∈ D2 , we choose w(u) so
that p(u) = 8;
8
• For each u ∈ C, if w assigns 2 to at least two of the four edges incident
to u and u has a neighbor v ∈ D2 , by construction, we have w(v) = 2
and w(uv) = 1, and u has exactly one neighbor x ∈ B, and w assigns
1 to exactly two of the four edges incident to u. If p(x) = 16, then
w(u) = 1 and p(u) = 4. If p(x) = 4, then w(v) = 1 and w(uv) = 2
(p(v) remains unchanged), and w(u) = 2 and p(u) = 16.
4
Digraphs
Only a few directed variations of the 1-2-3 Conjecture have been investigated
in the literature. Borowiecki, Grytczuk and Pilśniak [3] and Khatirinejad,
Naserasr, Newman, Seamone and Stevens [7] have independently studied a
variation where the color function for vertices is the sum of the weights of
incoming arcs (s− ) minus the sum of the weights of outgoing arcs (s+ ). We
call this potential-coloring. They have shown that if every arc is assigned
a list of size 2 from which its weight is chosen, it is possible to weight the
digraph in order to obtain a potential-coloring arc-weighing. In particular,
every digraph is potential-coloring 2-arc-weightable.
Baudon, Bensmail and Sopena have considered another version of the problem, where the color of vertices is s+ [1], which we will call out-sum-coloring.
They have proved that every digraph is out-sum-coloring 3-arc-weightable,
that some digraphs are not out-sum-coloring 2-arc-weightable, and that deciding whether a given digraph is out-sum-coloring 2-arc-weightable is an
NP-complete problem.
Those variations are not satisfying as, as shown by the results obtained for
them, they are not a proper transcription of the undirected problem to digraphs.
Here, we discuss another variation where every vertex has an in- and an
→ the out-color of u
out-color, and the coloring is proper if for every arc −
uv,
is distinct from the in-color of v. This variation, while close to the ones
described above, is a priori more complex, because modifying the color of a
vertex also modifies that of its neighbors.
We conjecture that similar results to the ones for the original problem could
be obtained for this directed variation.
Conjecture 4 (Directed 1-2-3 Conjecture). For every nice digraph D, we
have χaP (D) ≤ 3.
±
9
Figure 3: An acyclic bipartite digraph with χaP (D) = 3
±
This conjecture is supported by our results showing it holds for acyclic
digraphs and bipartite digraphs. One can easily convince oneself that for
some digraphs D, notably some bipartite digraphs, we have χaP (D) ≥ 3.
±
The bounds obtained for acyclic digraphs, acyclic tournaments and bipartite
digraphs, respectively 3, 2 and 3, are the best possible (Figure 3). We also
provide a first bound for general digraphs, by showing that every digraph is
sum-coloring 17-arc-weightable.
4.1
Acyclic digraphs
In this part, we focus on acyclic digraphs. We first show that acyclic tournaments are sum-coloring 2-arc-weightable, before proving that the conjecture
more generally holds for nice acyclic digraphs.
Lemma 5. For every acyclic tournament T of order s ≥ 3, we have χaP (T ) ≤
±
2.
Proof. Let T be an acyclic tournament of order s and w an arc-weighting of
→
T . We label the vertices v1 , ...vs so that i < j for every arc −
v−
i vj ∈ T . If s = 3,
→
−−→
the weighting is trivial, for example we can set w(−
v−
1 v2 ) = 2, and w(v1 v3 ) = 1
→
and w(−
v−
2 v3 ) = 1.
If s > 3, we weight arcs in the following way:
−→
−−−→
• For every odd i with 3 < i ≤ s, we set w(−
v−
i−2 vi ) = 1 and w(vi−1 vi ) = 1;
→
−−→
• For all other arcs −
v−
j vk ∈ T , we set w(vj vk ) = 2.
10
For all vertices u of T , we have s− (u) even. If s is odd, then for all vertices
→
u of T , we have s+ (u) odd or equal to 0, so s+ (u) 6= s− (v) for every arc −
uv
of T .
If s is even, then we have s+ (u) odd or equal to 0 for every vertex u of T
→ of T . In addition to
different from vs−1 , so s+ (u) 6= s− (v) for every arc −
uv
this, we have d+ (vs−1 ) = 1 so s+ (vs−1 ) = 2. vs is the only out-neighbor of
vs−1 , and we have s− (vs ) ≥ s, and s > 3, so s+ (vs−1 ) 6= s− (vs ).
Hence w is a sum-coloring 2-arc-weighting of T .
Using a similar idea, we prove that every nice acyclic graph is sum-coloring
3-arc-weightable.
Theorem 6. For every nice acyclic digraph D, we have χaP (D) ≤ 3.
±
Proof. Let D be a nice acyclic digraph. We will build a 3-arc-weighting w
of D in a way that guarantees that s− (u) is even for every u of D and that
maximizes the number of vertices u ∈ D with s+ (u) odd.
First, we build a set of vertices P in the following way: for every vertex v, we
choose the highest possible even number (possibly null) of vertices u ∈ D −P
→ ∈ D and add them to P . For all arcs −
→ with u ∈ P and v ∈ D,
such that −
uv
uv
→ to 1. For all arcs −
→ with u ∈ D − P and v ∈ D, we set w(−
→
we set w(−
uv)
uv
uv)
−
+
to 2. We now have s (u) even for all u of D and s (u) odd for all u of P .
→ ∈ D and
Then, let u be a vertex in D − P and x a vertex in D such that −
ux
s+ (u) = s− (x). If d− (x) = 1, then d+ (u) > 1 (or it would be a forbidden
pattern), and s+ (u) > 2 = s− (x), so we leave the weights unchanged. Else,
→ ∈ D. This vertex v is
x has at least one neighbor v such that u 6= v and −
vx
→ = 1. We set
not in D − P , else we could have added u and v to P , so w(−
vx)
→ to 3. s+ (v) is still odd and s− (x) still even, and now s+ (u) 6= s− (x).
w(−
vx)
Assume u has d+ neighbors v1 , ..., vd+ . For every vi , there is no x ∈ D − P
→ ∈ D, else we could add u and x to P . For every arc −
→, we
such that −
xv
uv
i
i
can either leave its weight unchanged or add 2. We can do it in a way that
leaves s+ (u) 6= s− (vi ) for every 1 ≥ i ≥ d+ . We repeat the procedure for all
vertices in D − P .
→ ∈ D.
We now have s+ (u) 6= s− (v) for every arc −
uv
4.2
Multipartite digraphs
When studying the 1-2-3 Conjecture and the related problems, one is essentially studying a specific coloring problem. Hence, it seems natural to
11
consider classes of graphs defined by their chromatic number, such as the
3-colorable graphs. Along those lines, we prove that the Directed 1-2-3 Conjecture holds for nice bipartite digraphs.
Lemma 7. For every nice bipartite digraph D, we have χaP (D) ≤ 3.
±
Proof. Let D = A ∪ B be a nice bipartite digraph and w an arc-weighting
of D. We can separate the edges of D into two nice bipartite graphs D1 and
→ ∈ D then x ∈ A and y ∈ B, and if −
→∈D
D2 with D = D1 ∪ D2 and if −
xy
xy
1
2
then x ∈ B and y ∈ A.
We can properly weight D1 and D2 the same way we would weight their
non-oriented underlying graphs [6]. We then repercutate the weigthings on
→ ∈ D with u ∈ A and v ∈ B, s+ (u)
D. There is no conflict because for −
uv
→ ∈ D with u ∈ B
and s− (v) were defined by the weighting of D1 , and for −
uv
and v ∈ A, s+ (u) and s− (v) were defined by the weighting of D2 .
After bipartite graphs, the logical continuation would be to consider 3colorable digraphs. However, for 3-colorable graphs, using the underlying
3-coloration of the undirected graph is not sufficient, as we do not know the
direction of arcs between two given parts. Therefore, it seems more natural
to consider digraphs with an oriented chromatic number 3 as equivalent to
non-oriented 3-colorable graphs. Digraphs with oriented chromatic number
k are k-colorable digraphs in which all arcs between a part Ai and a part Aj
share the same direction.
There are two types of digraphs with oriented chromatic number 3: acyclic
and circular digraphs. Below, we show that nice circular digraphs are sumcoloring 3-arc-weightable.
Definition 14 (Circular digraph).
A digraph D is circular if D is a multiS
→ ∈ D, if u ∈ A
partite digraph with D = 1≤i≤k Ai , where for every arc −
uv
i
then v ∈ Ai+1 mod k .
Corollary 7.1. For every nice circular digraph D, we have χaP (D) ≤ 3.
±
S
Proof. Let D be a nice circular digraph, D = 1≤i≤k Ai , withv ∈ Ai+1 mod k
if u ∈ Ai . In the same fashion as for a bipartite digraph, we can separate the
arcs of D into k bipartite graphs where all edges go in the same direction.
Again, we weight the edges of the underlying graphs and can repercutate the
weightings on D without conflict.
12
Lemma 8. For every nice digraph with oriented chromatic number 3, we
have χaP (D) ≤ 3.
±
Proof. Let D be a nice digraph with oriented chromatic number 3. D is
either circular, or acyclic. In both cases, we have χaP (D) ≤ 3, according to
±
Theorem 6 and Corollary 7.1.
We have shown that the Directed 1-2-3 Conjecture holds for nice acyclic
and bipartite digraphs, and that those bounds are the best possible, as shown
by the acyclic bipartite digraph in Figure 3.
4.3
General digraphs
Here, we show a weakening of the Directed 1-2-3 Conjecture, namely that
χaP (D) ≤ 17 for every nice digraph D.
±
Theorem 9. Every nice digraph is sum-coloring 17-arc-weightable.
The following proof, and especially the Initialization and Propagation
steps are adapted from the proof by Kalkowski, Karońshi and Pfender, showing that every nice graph is sum-coloring 5-edge-weightable [5]. The idea of
the proof is as follows: in order to obtain a proper weighting, we attribute
an arbitrary order to vertices; we weight all edges with a weight divisible
by 3 and assign to every vertex with at least one neighbor after it in the
order a set {w, w + 6} (with w ≡ 0 mod 3) for each direction (in-coming
and out-going arcs), disjoint from the sets of its neighbors; at the end, we
adjust the weights for vertices that have no neighbors after them in the order
by changing their weights modulo 3.
Proof. Let D be a nice digraph. If |D| ≤ 1 there are no arcs to weight. If
D is not connected, we can argue component wise. In addition to this, D
contains no forbidden pattern as described above. We may therefore assume
that D is connected and |D| ≥ 3.
We order the vertices in an arbitrary order v1 , ..., vn . We then divide the
vertices into categories: vi is a source+ (resp. source− ) if there is vj ∈ D
→
−−→
+
with i ≤ j ≤ n such that −
v−
(resp.
i vj ∈ D (resp. vj vi ∈ D), and a sink
−
sink ) otherwise.
For every source+ vi ∈ D, we will assign a set of out-colors Φ+ (vi ) =
{φ+ (vi ), φ+ (vi ) + 6} and for every source− vj a set of in-colors Φ− (vi ) =
13
→
+
−
{φ− (vi ), φ− (vi )+6}, so that for every arc −
v−
i vj ∈ D, we have Φ (vi )∩Φ (vj ) =
∅. We will weight D in three steps:
Initialization
→ ∈ D, we set w(−
→ = 9.
For every arc −
uv
uv)
We have s+ (v1 ) = 9 ∗ d+ (v1 ) and s− (v1 ) = 9 ∗ d− (v1 ). We choose φ+ (vi ) so
that s+ (v1 ) ∈ Φ+ (v1 ) and φ− (vi ) so that s− (v1 ) ∈ Φ− (v1 ).
Propagation
We treat vertices in order. Let 2 ≤ k ≤ n, we assume all vertices vi ∈ D, i ≤ k
have already been treated. We also have the following properties:
• For all i < k, if vi is a source+ , then s+ (vi ) ∈ Φ+ (vi ) and if vi is a
source− , then s− (vi ) ∈ Φ− (vi ).
→
−−→
• For all j > k, we have w(−
v−
j vk ) = 9 and w(vk vj ) = 9.
→
−−→
• For all i < k, we have w(−
v−
i vk ) ∈ {6, 9, 12} and w(vk vi ) ∈ {6, 9, 12}. If
→
−−→
−
−
−
w(−
v−
i vk ) = 6 then s (vi ) = φ (vi ), and if w(vi vk ) = 12 then s (vi ) =
→
−−→
−
+
φ− (vi ) + 6. If w(−
v−
k vi ) = 6 then s (vi ) = φ (vi ), and if w(vk vi ) = 12
then s+ (vi ) = φ+ (vi ) + 6.
We treat incoming and outgoing components separately. Here is the reasoning for the incoming arcs and color of vertex vk , outgoing arcs and color are
treated in the same fashion.
If vk is a sink − , we do not change anything and leave vk to be treated in
the last step. We do not need to define Φ+ (vi ) because vk has no neighbor
→
−
vj , j > k such that −
v−
j vk ∈ D (or it would not be a sink ).
−
−
→
If vk is a source− , for every arc vi vk ∈ D with i < k, we know vi is a source+
and we have defined Φ+ (vi ). We can either add or substract 6 from the
−−→
weight w(vi vk) while keeping s+ (vi ) in Φ+ (vi ). This way, we can generate
−−→
δ + 1 different values for s− (vk ), where δ is the number of arcs vi vk ∈ d with
→
i < k. Let l be the smallest j > k with −
v−
j vk ∈ D, vl has not been treated yet
−−→
and we allow ourselves to add or substract 3 from the weight w(vl vk). We
can now generate 2 ∗ δ + 3 different values for s− (vk ), we can choose φ− (vk )
so that:
• For all vi ∈ D with i < k, we have Φ+ (vi ) ∩ Φ− (vj ) = ∅.
• s− (vk ) ∈ Φ− (vk )
14
−−→
−−→
• If w(vl vk) = 6, then s− (vk ) = φ− (vk ). If w(vl vk) = 12, then s− (vk ) =
φ− (vk ) + 6.
We repeat this process for the incoming and outgoing color of all vertices.
Sinks weighting
At this point, all source+ vi have a set Φ+ (vi ) and s+ (vi ) ∈ Φ+ (vi ), and all
source− vi have a set Φ− (vi ) and s− (vi ) ∈ Φ− (vi ). Also, for every vertex
vi ∈ D, s+ (vi ) ≡ 0 mod 3 and s− (vi ) ≡ 0 mod 3. We still need to adjust
the sum-coloring of sinks to get a proper coloring. Once again, we will detail
the procedure for sink − vertices, the same applies to sink + vertices (a vertex
can be a sink for one or both directions).
We choose a sink − vi . We build the set S = {vj ∈ D, vj a sink − , ∃vk ∈
−−→
−−→
D, vk vi ∈ D & vk vj ∈ D}. S is the set of source− that share at least one
neighbor with vi . We then build the neighboring set N = {vk ∈ D, ∃vj ∈
−−→
S, vk vj ∈ D}. If S ∩ N 6= ∅, for every vertex vj ∈ S ∩ N , we split vj into
→ ∈ D, we have w(−
→ ≡ 0 mod 3 so for
vj+ ∈ N and vj− ∈ S. For every arc −
xy
xy)
+
every vertex u ∈ N , we have s (u) ≡ 0 mod 3 and for every vertex u ∈ S,
we have s− (u) ≡ 0 mod 3.
If |S| = 1 and |N | = 1, let u be the vertex in S and v the vertex in N .
→ ∈ D (or it would be a forbidden
Then there is a vertex x ∈ D \ (S ∪ N ), −
vx
→ and s+ (v) > 3 ∗ (d+ (v) − 1) + w(−
→ with
pattern). We have s− (u) = w(−
vu)
vu)
−
→
+
+
−
d (v) > 1 so we leave w(vu) unchanged and s (v) > s (u).
If |S| = 1 and |N | > 1, let u be the vertex in S and x and y be two vertices
→ and w(−
→ We have s+ (u) ≡ 2 mod 3 and
in N . We add 1 to w(−
xu)
yu).
s+ (x) ≡ s+ (y) ≡ 1 mod 3 and for every vertex v ∈ N \ {x, y}, we have
s+ (v) ≡ 0 mod 3. This does not cause any conflict with the rest fo the
→ ∈ D.
graph as for every vertex v ∈ N , there is no sink − z ∈ D \ S with −
vz
If |S| > 1, by construction there is at least one vertex in N with out-degree
at least 2. Let G be the non-oriented sub-graph underlying D ∩ (S ∪ N ). G
is bipartite, with S ∩ N = ∅. Let wG be the weighting function and sG the
vertex color function in G. We first set wG (uv) = 0 for every edge (uv) ∈ G.
We will then process G in four steps in order to reach sG (u) ≡ 2 mod 3 for
every vertex u ∈ N and sG (u) ∈ {0, 1} mod 3 for every vertex u ∈ S and
sG (u) ∈ {0, 1, 2} for every vertex u ∈ N :
• We choose two vertices u ∈ N with sG (u) ∈ {0, 1} mod 3 and v ∈ N
with sG (v) ∈ {0, 1} mod 3, and a path from u to v. We set k ∈ {0, 1}
15
such that sG (u) ≡ k mod 3. Add 2 − k to the first edge of this path,
1 + k to the second, 2 − k to the third... We now have sG (u) ≡ 2
mod 3. For every x ∈ G with x ∈
/ {u, v}, the value sG (x) mod 3 has
not changed. We can repeat this until only one vertex y ∈ N with
sG (y) ∈ {0, 1} mod 3 is left, and d(y) ≥ 2. Note that for every vertex
u of S, we have sG (u) ≡ 0 mod 3.
• If sG (y) ≡ 0 mod 3, choose two edges (x, y) ∈ G and add one to
wG (x, y). If sG (y) ≡ 1 mod 3, choose one edge (x, y) ∈ G and add one
to wG (x, y). We now have sG (u) ≡ 2 mod 3 for every vertex u ∈ N
and sG (u) ∈ {0, 1} mod 3 for every vertex u ∈ S, with sG (u) ≡ 1
mod 3 for at most two vertices u ∈ S.
• For every edge (u, v) ∈ G, we choose k ∈ {0, 1, 2} with wG (u, v) ≡ k
mod 3, and we set wG (u, v) = k.
• For every vertex u ∈ N , if sG (u) ≥ 3, we substract 3 from as many
edges as needed to get 0 ≤ sG (u) ≤ 3. We need to substract at most
once by edge as the initial weight of edges is smaller than 3, therefore,
the final weights on the edges on G belong to {−2, −1, 0, 1, 2}.
We reverberate the weight of G on D by adding, for every edge (u, v) ∈ G
the (possibly negative) weight wG (u, v) to the weight of the corresponding
→ or −
→
arc −
uv
vu.
Repeat the process for sink + vertices. All neighboring vertices colors are
differentiated either by their rest in the division by 3 or because they belong
to disjoint Φ sets.
5
Conclusion
The 1-2-3 Conjecture and associated variations provide numerous optimization problems. In this report, we focused specifically on two variations: the
multiplicative total-weighting variation and a new directed variation.
The Multiplicative 1-2 Conjecture has been shown to hold for 3-colorable
graph, and we tried to extend it to 4-colorable graphs. We showed that
this conjecture holds for 4-regular graphs, meaning every 4-regular graph is
product-coloring 2-total-weightable. This result could maybe be extended to
k-regular graphs with k > 4. In regards to 4-colorable graphs, we could not
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show that the conjecture holds for them, but it would be interesting to look
at sub-classes such as planar graphs.
We also considered a new directed variation, where a digraph D is properly
→ of D, the sum of the weights on
sum-coloring arc-weighted if for every arc −
uv
incoming arcs of a vertex u is different from the sum of the weights on outgoing arcs of v. In relation with the original 1-2-3 Conjecture, we conjectured
that nice digraphs are sum-coloring 3-arc-weightable. We proved that this
conjecture holds for a few classes of graphs, namely nice bipartite digraphs,
nice digraphs with oriented chromatic number 3 and nice acyclic digraphs,
and that this result cannot be improved for those classes of graphs.
We then provided a more relaxed bound for sum-coloring k-arc-weighting of
every nice digraph, namely k = 17. In the proof for this relaxed bound,
it appears that instead of alleviating constraints on the coloring, dividing
the color of vertices into two colors is making the problem more complex.
Indeed, the algorithm used in the proof gives a bound of k = 5 for nonoriented graphs and k = 17 for digraphs for sum-coloring k-arc-weighting.
Some ground might be won by changing the way the last step is adapted
from the original algorithm, but the original algorithm cannot be used as it
is.
To get a better grasp of the particularities of this directed variation, it would
be interesting to study the directed equivalent for other classes of graphs for
which the 1-2-3 Conjecture has already been proved, in particular 3-colorable
graphs. Another logical continuation would be to try and characterize digraphs which are sum-coloring 2-arc-weightable.
5.1
Aknowledgment
Many thanks to Julien Bensmail, for scientific and methodological supervision and support during this internship. I would also like to thank the MC2
team for being so welcoming, and in particular Aurélie Lagoutte and Ignacio
García-Marco for sharing their office and their jokes with me.
References
[1] O. Baudon, J. Bensmail, and É. Sopena. An oriented version of the 1-23 conjecture. Discussiones Mathematicae Graph Theory, 35(1):141–156,
2015.
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[2] J. Bensmail. Partitions and decompositions of graphs. PhD thesis, Université de Bordeaux, 2014.
[3] M. Borowiecki, J. Grytczuk, and M. Pilśniak. Coloring chip configurations on graphs and digraphs. Information Processing Letters, 112(1):1–
4, 2012.
[4] M. Kalkowski. A note on the 1, 2-conjecture. submitted (also in Ph. D.
Thesis, 2009), 2009.
[5] M. Kalkowski, M. Karoński, and F. Pfender. Vertex-coloring edgeweightings: towards the 1-2-3-conjecture. Journal of Combinatorial
Theory, Series B, 100(3):347–349, 2010.
[6] M. Karoński, T. Łuczak, and A. Thomason. Edge weights and vertex
colours. Journal of Combinatorial Theory, Series B, 91(1):151 – 157,
2004.
[7] M. Khatirinejad, R. Naserasr, M. Newman, B. Seamone, and B. Stevens.
Digraphs are 2-weight choosable. the electronic journal of combinatorics,
18(1):P21, 2011.
[8] J. Przybyło and M. Woźniak. On a 1, 2 conjecture. Discrete Mathematics
& Theoretical Computer Science, 12(1):101–108, 2010.
[9] B. Seamone. The 1-2-3 Conjecture and related problems: a survey.
ArXiv e-prints, Nov. 2012.
[10] J. Skowronek-Kaziów. Conjecture—the multiplicative version. Information Processing Letters, 107(3–4):93 – 95, 2008.
[11] J. Skowronek-Kaziów. Multiplicative vertex-colouring weightings of
graphs. Information Processing Letters, 112(5):191–194, 2012.
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A
Institutional and social context
I did my internship in the MC2 (Modèles de calcul, Complexité, Combinatoire) team in the Laboratoire de l’Informatique du Parallèlisme (LIP) lab
at the École Normale Supérieure de Lyon. With more than a hundred researchers and students, research at the LIP covers a wide range of key topics
in computer and information sciences. MC2 main themes of research are
combinatorics, complexity theory, and algorithmic of discrete structures and
algebraic problems.
During my time with the MC2 team, I interacted with researchers from
diverse nationalities and at every stage of their career, from permanent researchers to post-docs, to PhD students. I shared an office with two post-docs
and a PhD. student, who worked respectively on algebraic complexity, probabilistic methods applied to graphs, and graph coloring problems.
I attended the team workshops which took place during my stay, both on
word algorithmic, as well as a seminary at the LIRIS, a collaborating lab,
where Aurélie, the PhD. student in my office, presented her work. I was also
able to discover research done in the other teams of the lab during a poster
session organized by PhD students.
In the office and in the team, the atmosphere was really friendly and I really
enjoyed those six weeks.
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B
Additionnal definitions
Definition 15 (Degree). The degree of a vertex is the number of its neighbors.
Definition 16 (Cycle). A cycle is a simple graph in which every vertex has
exactly two neighbors.
Definition 17 (Complete graph). A complete graph is a simple graph in
which each pair of vertices share one edge.
Definition 18 (K2 ). A graph K2 is a complete graph with two vertices.
Definition 19 (Tournament). A tournament is a simple digraph in which
each pair of vertices share one and only one arc.
Definition 20 (Colorability). A graph is k-colorable if one can attribute a
color out of k colors to every vertex of the graph so that every two neighbor
vertices have distinct colors.
The chromatic number of a graph is the smallest integer k for which the
graph is k-colorable.
Definition 21 (Bipartite). A bipartite graph is a 2-colorable graph.
Theorem 10 (Brook’s Theorem). For any connected undirected graph G
with maximum degree ∆, the chromatic numbre of G is at most ∆, unless G
is a complete graph or an odd cycle, in which case the chromatic number is
∆ + 1.
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