Injective coloring of planar graphs with girth 6

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 χi(G) be the injective chromatic number of a graph G. In this paper, we investigate the injective coloring of planar graphs with girth 6. We improve some results of Borodin and Ivanova (2011) [1], Doyon et al. (2010) [4] and Li and Xu (2012) [6] by showing that if G is a planar graph with girth at least 6, then (1) χi(G)≤Δ+3; (2) χi(G)≤Δ+2 if Δ≥9; (3) χi(G)≤Δ+1 if Δ≥17.