List injective coloring of planar graphs with disjoint 5−-cycles

Abstract

An injective [Formula: see text]-coloring of a graph [Formula: see text] is called injective if any two vertices joined by a path of length two get different colors. A graph [Formula: see text] is injectively [Formula: see text]-choosable if for any color list [Formula: see text] of admissible colors on [Formula: see text] of size [Formula: see text] it allows an injective coloring [Formula: see text] such that [Formula: see text] whenever [Formula: see text]. Let [Formula: see text], [Formula: see text] denote the injective chromatic number and injective choosability number of [Formula: see text], respectively. Let [Formula: see text] be a plane with disjoint [Formula: see text]-cycles and maximum degree [Formula: see text]. We show that [Formula: see text] if [Formula: see text], then [Formula: see text]; [Formula: see text] if [Formula: see text], then [Formula: see text].