Injective colorings of planar graphs with few colors

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. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥19 and maximum degree ∆ are injectively ∆-colorable. We also show that all planar graphs of girth ≥10 are injectively (∆ + 1)-colorable, ∆ + 4 colors are sufficient for planar graphs of girth ≥ 5 if ∆ is large enough, and that subcubic planar graphs of girth ≥7 are injectively 5-colorable.