List injective colorings of planar graphs

AbstractA vertex coloring of a graph G 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 φ such that φ(v)∈L(v) whenever v∈V(G). The least k for which G is injectively k-choosable is denoted by χil(G).Note that χil≥Δ for every graph with maximum degree Δ. For planar graphs with girth g, Bu et al. (2009) [15] proved that χil=Δ if Δ≥71 and g≥7, which we strengthen here to Δ≥16. On the other hand, there exist planar graphs with g=6 and χil=Δ+1 for any Δ≥2. Cranston et al. (submitted for publication) [16] proved that χil≤Δ+1 if g≥9 and Δ≥4. We prove that each planar graph with g≥6 and Δ≥24 has χil≤Δ+1