Injective coloring of plane graphs with girth 5

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. Let G be a plane graph with g ( G ) ≥ 5 and χ i ( G ) be the injective chromatic number of G . In this paper, we improve some known results by proving that χ i ( G ) ≤ Δ + 3 if Δ ( G ) ≥ 35 and χ i ( G ) ≤ Δ + 6 for arbitrary Δ ( G ) .