# Injective colorings of sparse graphs

#### Abstract

Let mad(G) denote the maximum average degree (over all subgraphs) of G and let chi(i)(G) denote the injective chromatic number of G. We prove that if mad(G) < 5/2, then chi(i)(C) <= Delta(G) + 1; and if mad(G) < 42/19, then chi(i)(C) = Delta(C). Suppose that G is a planar graph with girth g(G) and Delta(G) >= 4. We prove that if chi(i)(G) >= 9, then chi(i)(G) <= Delta(G) + 1; similarly, if g(G) >= 13, then chi(i)(C) = Delta(C). (C) 2010 Elsevier B.V. All rights reserved.

*This paper has been withdrawn.*