List injective coloring of planar graphs

Abstract

An injective coloring is a vertex coloring (not necessarily proper) such that any two vertices sharing a common neighbor receive distinct colors. A graph G is called injectively k-choosable, if for any color list L with admissible colors on V(G) of size k, there is an injective coloring φ such that φ(v)∈L(v) whenever v∈V(G). The list injective chromatic number, denoted by χil(G), is the least k for which G is injectively k-choosable. We focus on the study of list injective coloring on planar graphs which has disjoint 5−-cycles and show that χil(G)≤Δ+3 if Δ≥18 and χil(G)≤Δ+4 if Δ≥12.