Graph multicoloring reduction methods and application to McDiarmid-Reed's Conjecture

A $(a,b)$-coloring of a graph $G$ associates to each vertex a set of $b$ colors from a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoints. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, we prove necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices. Other more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice. Computations on millions of such graphs generated randomly show that our tools allow to find (in linear time) a $(9,4)$-coloring for each of them. Although there remain few graphs for which our tools are not sufficient for finding a $(9,4)$-coloring, we believe that pursuing our method can lead to a solution of the conjecture of McDiarmid-Reed.Comment: 27 page

Simpler multicoloring of triangle-free hexagonal graphs

International audienceGiven a graph G and a demand function p:V(G)→N, a proper n-[p]coloring is a mapping f:V(G)→2{1,...,n} such that |f(v)|≥p(v) for every vertex v∈V(G) and f(v)∩f(u)=0̸ for any two adjacent vertices u and v. The least integer n for which a proper n-[p]coloring exists, χp(G), is called multichromatic number of G. Finding multichromatic number of induced subgraphs of the triangular lattice (called hexagonal graphs) has applications in cellular networks. Weighted clique number of a graph G, ωp(G), is the maximum weight of a clique in G, where the weight of a clique is the total demand of its vertices. McDiarmid and Reed (2000) [8] conjectured that χp(G)≤(9/8)ωp(G)+C for triangle-free hexagonal graphs, where C is some absolute constant. In this article, we provide an algorithm to find a 7-[3]coloring of triangle-free hexagonal graphs (that is, when p(v)=3 for all v∈V(G)), which implies that χp(G)≤(7/6)ωp(G)+C. Our result constitutes a shorter alternative to the inductive proof of Havet (2001) [5] and improves the short proof of Sudeep and Vishwanathan (2005) [17], who proved the existence of a 14-[6]coloring. (It has to be noted, however, that our proof makes use of the 4-color theorem.) All steps of our algorithm take time linear in |V(G)|, except for the 4-coloring of an auxiliary planar graph. The new techniques may shed some light on the conjecture of McDiarmid and Reed (2000) [8]