Injective Δ+2 Coloring of Planar Graph Without Short Cycles

A coloring of graph G is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number χ_i(G) of G is the least integer k such that G has an injective k-coloring. In this paper, we prove that (1) if G is a planar graph with girth g ≥ 6 and maximum degree Δ ≥ 7, then χ_i(G) ≤ Δ + 2; (2) if G is a planar graph with Δ ≥ 24 and without 3,4,7-cycles, then χ_i(G) ≤ Δ + 2.