Injective choosability of planar graphs of girth five and six

An injective k-coloring of a graph G is an assignment of k colors to V (G) such that vertices having a common neighbor receive distinct colors. we study the list version of injective colorings of planar graphs. Let chi(l)(i)(G) and mad(G) be the injective choosability number and the maximum average degree of G, respectively. It is proved that (1) for each graph G with mad(G) < 10/3, chi(l)(i)(G) <= Delta(G) + 4 if Delta(G) >= 30 (this conditionally improves some results of Doyon et al. (2010) [9] and Luzar et al. (2009)[11]). chi(l)(i)(G) <= Delta(G) + 5 if Delta(G) >= 18, and chi(l)(i)(G) <= Delta(G) + 6 if Delta(G) >= 14; (2) chi(l)(i)(G) <= Delta(G) + 2 if mad(G) < 3 and Delta(G) >= 12 (this conditionally improves a result of Doyon et al. (2010) [9]). (C) 2011 Elsevier B.V. All rights reserved.