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 GG be a plane graph with g(G)≥5g(G)≥5 and χi(G)χi(G) be the injective chromatic number of GG. In this paper, we improve some known results by proving that χi(G)≤Δ+3χi(G)≤Δ+3 if Δ(G)≥35Δ(G)≥35 and χi(G)≤Δ+6χi(G)≤Δ+6 for arbitrary Δ(G)Δ(G).