Wegner's Conjecture on 2-distance coloring for planar graphs

A 2 distance k-coloring of a graph G is a function f:V(G)→{1,2,…,k} such that |f(u)−f(v)|≥1 if 1≤d(u,v)≤2, where d(u,v) is the distance between the two vertices u and v. The 2-distance chromatic number of G, written χ2(G), is the minimum k such that G has such a coloring. In this paper, we show that for planar graphs G, χ2(G)≤5Δ(G)−9 if 7≤Δ(G)≤8, which improves a result due to Zhu and Bu (2018) [9].