List injective colorings of planar graphs

A vertex coloring of a graph C is called injective if any two vertices joined by a path of length two get different colors. A graph G is injectively k-choosable if any list L of admissible colors on V(G) of size k allows an injective coloring phi such that phi(nu) is an element of L(nu) whenever nu is an element of V(G). The least k for which G is injectively k-choosable is denoted by chi(l)(j)(G). Note that chi(l)(j) >= Delta for every graph with maximum degree Delta. For planar graphs with girth g, Bu et al. (2009) [15] proved that chi(l)(j) = Delta if Delta >= 71 and g >= 7, which we strengthen here to Delta >= 16. On the other hand, there exist planar graphs with g = 6 and chi(l)(j) = Delta + 1 for any Delta >= 2. Cranston et al. (submitted for publication) [16] proved that chi(l)(j) < Delta + 1 if g >= 9 and Delta >= 4. We prove that each planar graph with g >= 6 and Delta >= 24 has chi(l)(j) <= Delta + 1. (c) 2010 Elsevier B.V. All rights reserved.