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 χ i l ( 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 , χ i l ( 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]), χ i l ( G ) ≤ Δ ( G ) + 5 if Δ ( G ) ≥ 18 , and χ i l ( G ) ≤ Δ ( G ) + 6 if Δ ( G ) ≥ 14 ; (2) χ i l ( G ) ≤ Δ ( G ) + 2 if mad ( G ) < 3 and Δ ( G ) ≥ 12 (this conditionally improves a result of Doyon et al. (2010) [9]).