List injective colorings of planar graphs

A 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 χ i l ( G ) . hat χ i l ≥ Δ for every graph with maximum degree Δ . For planar graphs with girth g , Bu et al. (2009) [15] proved that χ i l = 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 χ i l = Δ + 1 for any Δ ≥ 2 . Cranston et al. (submitted for publication) [16] proved that χ i l ≤ Δ + 1 if g ≥ 9 and Δ ≥ 4 . We prove that each planar graph with g ≥ 6 and Δ ≥ 24 has χ i l ≤ Δ + 1 .