INJECTIVE COLORING OF PLANAR GRAPHS WITH GIRTH 7

Abstract

An injective k-coloring of a graph G is an assignment of k colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors, and χ i (G) is the injective chromatic number of G. Dimitrov et al. proved χ i (G) ≤ Δ(G) + 2 for a planar graph G with g(G) ≥ 7. In this paper, we show that if G is a planar graph with g(G) ≥ 7 and Δ(G) ≥ 7, then χ i (G) ≤ Δ(G) + 1.