Abstract
Given 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].