A linear time algorithm for 7-[3]coloring triangle-free hexagonal graphs

Given a graph G, a proper n − [p]-coloring is a mapping f : V (G) → 2{1,...,n} such that |f(v)| = p for any vertex v ∈ V (G) and f(v) ∩ f(u) = ∅ for any pair of adjacent vertices u and v. n − [p]-coloring is closely related to multicoloring. Finding multicoloring of induced subgraphs of the triangular lattice (called hexagonal graphs) has important applications in cellular networks. In this article we provide an algorithm to find a 7-[3]-coloring of triangle-free hexagonal graphs in linear time, which solves the open problem stated in [10] and improves the result of Sudeep and Vishwanathan [11], who proved the existence of a 14-[6]coloring. ∗This work was supported by grant N206 017 32/2452 for years 2007-2010