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 Delta are injectively Delta-colorable. We also show that all planar graphs of girth >= 10 are injectively (Delta + 1)-colorable, that Delta + 4 colors are sufficient for planar graphs of girth >= 5 if Delta is large enough, and that subcubic planar graphs of girth >= 7 are injectively 5-colorable. (C) 2008 Elsevier B.V. All rights reserved.