Injective choosability of planar graphs of girth five and six

AbstractAn 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 χil(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)<103, χil(G)≤Δ(G)+4 if Δ(G)≥30 (this conditionally improves some results of Doyon et al. (2010) [9] and Lužar et al. (2009) [11]), χil(G)≤Δ(G)+5 if Δ(G)≥18, and χil(G)≤Δ(G)+6 if Δ(G)≥14; (2) χil(G)≤Δ(G)+2 if mad(G)