On $$\alpha $$-Vertex Choosability of Graphs

Abstract

A connected, simple graph G with vertex set $$V(G)=\{1,2,\ldots ,n\}$$ is said to be vertex (n, k)-choosable, if there exists a collection of subsets $$\left\{ S_k(v)\subseteq V(G): v\in V\right\} $$ of cardinality k, such that $$S_k(u)\cap S_k(v)=\emptyset $$ for all $$uv\in E(G)$$ , where k is a positive integer less than n. The maximum value of such k is called the vertex choice number of G. In this paper, we introduce the notion of $$\alpha $$ - choosability of graphs in terms of their vertex (n, k)-choice number and initiate a study on the structural characteristics of $$\alpha $$ -choosable graphs.