Injectively Delta choosable graphs

Abstract

An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors. A graph G is said to be injectively k-choosable if any list L(v) of size at least k for every vertex v allows an injective coloring (v) such that (v) 2 L(v) for every v 2 V (G). The least k for which G is injectively k-choosable is the injective choosability number of G, denoted by l i(G). In this paper, we obtain new sufficient conditions to be l i(G) = △(G). Maximum average degree, mad(G), is defined by mad(G) = max{2e(H)/n(H) : H is a subgraph of G}. We prove that if mad(G) < [수식], then l i(G) = △(G) where k = △(G) and △(G) ≥ 6. In addition, when △(G) = 5 we prove that l i(G) = △(G) if mad(G) < 17 7 , and when △(G) = 4 we prove that l i(G) = △(G) if mad(G) < 7/3 . These results generalize some of previous results in [1, 4].