INJECTIVELY DELTA CHOOSABLE GRAPHS

Abstract

Abstract. An injective coloring of a graph G is an assignment of colorsto the vertices of G so that any two vertices with a common neighborreceive distinct colors. A graph G is said to be injectively k-choosableif any list L(v) of size at least k for every vertex v allows an injectivecoloring φ(v) such that φ(v) ∈ L(v) for every v ∈ V (G). The least k forwhich G is injectively k-choosable is the injective choosability number ofG, denoted by χ li (G). In this paper, we obtain new sufficient conditionsto be χ li (G) = ∆(G). Maximum average degree, mad(G), is defined bymad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that ifmad(G) < 8k−33k , then χ li (G) = ∆(G) where k = ∆(G) and ∆(G) ≥ 6. Inaddition, when ∆(G) = 5 we prove that χ li (G) = ∆(G) if mad(G) < 177 ,and when ∆(G) = 4 we prove that χ li (G) = ∆(G) if mad(G) < 73 . Theseresults generalize some of previous results in [1, 4]. 1. IntroductionAll graphsconsidered in this paper aresimple, finite, and undirected. We useV (G),E(G) and ∆(G) to denote the vertex set, the edge set and the maximumdegree of G, respectively. Here we introduce some notation. A k-vertex is avertex of degree k, and a k