List injective coloring of planar graphs with girth g≥6g≥6

A vertex coloring of a graph GG is called injective if any two vertices with a common neighbor receive distinct colors. A graph GG is injectively kk-choosable if any list LL of admissible colors on V(G)V(G) of size kk allows an injective coloring φφ such that φ(v)∈L(v)φ(v)∈L(v) whenever v∈V(G)v∈V(G). The least kk for which GG is injectively kk-choosable is denoted by χil(G). In this paper, we show that if GG is a planar graph with girth g≥6g≥6, then χil(G)≤Δ+3.