Section: New Results

Some Results in Graph Theory

Participants : Miguel Couceiro, 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 vertexes, 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 vertexes, 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 $n$ vertexes, there is a color $c$ and a set $X$ of 4-vertexes such that at least $2n/3$ vertexes 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.

Algorithmic Graph Theory and Clustering

Since September 2013, Mario Valencia has obtained a two years 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. The principal studied problem is known as the Cluster Deletion Problem: given a graph with real non negative edge weights, partition the vertexes into clusters (in this case cliques) in order to minimize the total weight of edges out of the clusters. Two papers were submitted to journals in 2014. In [94] , we discovered a one-to-one correspondence between potential solutions of the cluster deletion problem and the minimum sum coloring problem, and use it to obtain a polynomial time algorithm to solve the cluster deletion problem in a special family of graphs called $P_4$-reducible graphs.

In [95] , we studied the complexity of the cluster deletion problem on subclasses of chordal graphs and cographs. In particular, it is shown that the cluster deletion problem is NP-hard for unweighted chordal graphs and weighted cographs. Some polynomial-time solvable cases are also identified.

Moreover, the paper "b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs", has been accepted for publication in the journal Algorithmica [1] .

Structural and Algebraic Graph Theory

We have also worked on the following topics. Golumbic, Lipshteyn and Stern proved that every graph can be represented as the edge intersection graph of paths on a grid, i.e., one can associate to each vertex of the graph a nontrivial path on a grid such that two vertexes are adjacent if and only if the corresponding paths share at least one edge of the grid. For a non-negative integer $k$, $B_k$-EPG graphs are defined as graphs admitting a model in which each path has at most $k$ bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is $B_4$-EPG, by embedding the circle into a rectangle of the grid. We proved also that every circular-arc graph is $B_3$-EPG (paper submitted).

We have studied the $k$-tuple chromatic number of the Cartesian product of two graphs $G$ and $H$ in [96] . We have shown that there exist graphs $G$ and $H$ such that ${\chi }_k(G\square H)>max\{{\chi }_k\left(G\right),{\chi }_k\left(H\right)\}$ for $k\ge 2$. Moreover, we have also shown that there exist graph families such that, for any $k\ge 1$, the $k$-tuple chromatic number of their Cartesian product is equal to the maximum $k$-tuple chromatic number of its factors.