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 χ i ( 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) χ i ( G ) ≤ Δ + 3 ; (2) χ i ( G ) ≤ Δ + 2 if Δ ≥ 9 ; (3) χ i ( G ) ≤ Δ + 1 if Δ ≥ 17 .