Injective Coloring of Planar Graphs with girth 7

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 χi(G) is the injective chromatic number of G. Dimitrov et al. proved χi(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 χi(G) ≤ Δ(G) + 1.