Section: New Results

Some results in graph theory

Participants : Amedeo Napoli, Chedy Raïssi, Jean-Sébastien Sereni, Mario Valencia.

Keywords:

graph theory, extremal graph theory, coloring, clustering

Structural and extremal graph theory

Regarding graph coloring, a conjecture of Gera, Okamoto, Rasmussen and Zhang on set coloring was solved. A set coloring of a graph $G=(V,E)$ is a function $c:V\to \{1,...,k\}$ such that whenever $u$ and $v$ are adjacent vertices, it holds that $\left\{c\right(x):x\text{neighbor}\text{of}u\}\ne \left\{c\right(x):x\text{neighbor}\text{of}v\}$. In other words, there must be at least one neighbor of $u$ that has a color not assigned to a neighbor of $v$, or vice-versa. The smallest $k$ such that $G$ admits a set coloring is the set coloring number ${\chi }_s\left(G\right)$. We confirmed the conjecture by proving that ${\chi }_s\left(G\right)\ge \lceil {log}_2\chi \left(G\right)\rceil +1$, where $\chi \left(G\right)$ is the (usual) chromatic number of $G$. This bound is tight.

Works have been started on a 12-year-old conjecture by Heckman and Thomas about the fractional chromatic number of graphs with no triangles and maximum degree at most 3. This conjecture is actually a natural generalization of a fact established by Staton in 1979. Heckman and Thomas posits that in every graph with no triangles, maximum degree at most 3 and arbitrary weights on the vertices, there exists an independent set of weight at least $5/14$ times the total weight of the graph.

Regarding extremal graph theory, two results have been obtained. The first one deals with permutation snarks, while the second one reads as follows.

For every 3-coloring of the edges of the complete graph on $n$ vertices, there is a color $c$ and a set $X$ of 4-vertices such that at least $2n/3$ vertices are linked to a vertex in $X$ by an edge of color $c$.

This theorem is motivated by a conjecture of Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp from 1989, which asserts that $X$ can be of size 3 only. However, they were only able to prove that $X$ can be of size 22. Recently, Rahil Baber and John Talbot managed to build upon our work in a very nice article: adding a new idea to our argument, they managed to confirm the conjecture.

Graph theory and other fields

Interactions of graph theory with other topics (theoretical computer science, number theory, group theory, sociology and chemistry) have been considered. Most of them are still in progress and some are published. For instance, regarding distributed computing, the purpose of our work was to question the global knowledge each node is assumed to start with in many distributed algorithms (both deterministic and randomized). More precisely, numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Noticeable examples are the MIS algorithms and the $(\Delta +1)$-coloring algorithms. Unfortunately, most known local algorithms are non-uniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the number of nodes $n$. Our work provides a rather general method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art non-uniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem.

Other aspects on graph coloring and clustering

Since September 2013, Mario Valencia has obtained a one year invitation (namely Inria “Délégation”) for working at Inria Nancy – Grand Est, in the Orpailleur team, on graph theoretical aspects and data clustering. This research work consists in studying the modular decomposition techniques on the threshold graphs issues of the clustering process. More precisely, this study relies on families of graphs having a “good” decomposition as cographs and chordal graphs, and then, and on the analysis of the adaptation of these two families of graphs within a clustering activity.

Other research dimensions are dealing with algorithmic aspects of some variations of the classical graph coloring problem.

• Packing colorings of graphs where we need to color the vertices of a graph in such a way that vertices having a same color $c$ should be at a distance at least equal to $c+1$ in the graph. With P. Torres, a postdoc student, we have obtained some upper bounds for the packing chromatic number of hypercubes graphs of dimension $n$, denoted by $Q_n$, and we have computed exactly this parameter for this family of graph for $n=6,7,8$, extending previous results known for $n=2,3,4,5$ [35] .

• $(k,i)$-coloring of graphs, which is a generalization of a $k$-tuple coloring of graphs: given positive integers $k$ and $i$, we want to affect to each vertex a $k$-set of colors such that the intersection of the $k$-sets affected to adjacent vertices has cardinality at most equal to $i$. With F. Bonomo, I. Koch, and G. Duran, we have found a linear time algorithm for this problem on cycles and cacti graphs. Moreover, we have obtained an interesting equivalence between this problem on complete graphs and a problem on weighted binary codes.

• b-coloring of graphs, where we need to color the vertices of a graph in such a way that in each color class $j$ there exists at least one vertex $x_j$ adjacent to at least one vertex in all the other color classes. The goal of this problem is to maximize the number of colors under such a constraint (i.e. the b-chromatic number of a graph). With F. Bonomo, O. Schaudt and M. Stein, we have shown that b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs [77] .