List Injective Coloring of Planar Graphs

Abstract

A k-coloring of a graph G is a mapping c: V(G) → {1, 2, ⋯, k}. The coloring c is called injective if any two vertices have a common neighbor get distinct colors. A graph G is injectively k-choosable if for any color list L of admissible colors on V(G) of size k allows an injective coloring φ such that φ(v) ∈ L(v) for each v ∈ V(G). Let χi(G), χ ^l_i (G) denote the injective chromatic number and injective choosability number of G, respectively. In this paper, we show that χ ^l_i (G) ≤ Δ + 4 if Δ ≥ 22 and χ ^l_i (G) ≤ Δ + 5 if Δ ≥ 15, where G is a triangle-free planar graph and without intersecting 4-cycles.