کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6423541 | 1342400 | 2012 | 7 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Simpler multicoloring of triangle-free hexagonal graphs Simpler multicoloring of triangle-free hexagonal graphs](/preview/png/6423541.png)
Given a graph G and a demand function p:V(G)âN, a proper n-[p]coloring is a mapping f:V(G)â2{1,â¦,n} such that |f(v)|â¥p(v) for every vertex vâV(G) and f(v)â©f(u)=0̸ for any two adjacent vertices u and v. The least integer n for which a proper n-[p]coloring exists, Ïp(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, Ïp(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 Ïp(G)â¤(9/8)Ïp(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 vâV(G)), which implies that Ïp(G)â¤(7/6)Ïp(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].
Journal: Discrete Mathematics - Volume 312, Issue 1, 6 January 2012, Pages 181-187