Minimum 2-distance coloring of planar graphs and channel assignment

Abstract

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. $\chi _{2}(G)$=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree $\Delta$, $\chi _2(G) \le 7$ if $\Delta \le 3$, $\chi _2(G) \le \Delta +5$ if $4\le \Delta \le 7$ and $\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1$ if $\Delta \ge 8$. In this paper, we prove that: (1) If G is a planar graph with maximum degree $\Delta \le 5$, then $\chi _{2}(G)\le 20$; (2) If G is a planar graph with maximum degree $\Delta \ge 6$, then $\chi _{2}(G)\le 5\Delta -7$.