The Result on $(3,1)\ast$-Choosability of Graphs of Nonnegative Characteristic without $4$-Cycles and Intersecting Triangles

Abstract

A graph $G$ is called $(k,d{)}^{\ast }$-choosable if for every list assignment $L$ satisfying $|L(v)|\ge k$ for all $v\in V(G)$, there is an $L$-coloring of $G$ such that each vertex of $G$ has at most $d$ neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without $4$-cycles and intersecting triangles is $(3,1{)}^{\ast }$-choosable.