Injective coloring of planar graphs

Abstract

An injective k-coloring of a graph G is a mapping f:V(G)→{1,2,…,k} such that for any two vertices v1,v2∈V(G), f(v1)≠f(v2) if N(v1)∩N(v2)≠∅. The injective chromatic number of a graph G, denoted by χi(G), is the smallest integer k such that G has an injective k-coloring. In this paper, we prove that for a Halin graph G, χi(G)≤Δ(G)+2. Moreover, χi(G)≤Δ(G)+1 if Δ(G)≥6. Also, we show that for a triangle-free planar graph G without intersecting 4-cycles, χi(G)≤Δ(G)+6 if Δ(G)≥20.