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

Abstract

A graph \(G\) is called \((k, d)^*\)-choosable if for every list assignment \(L\) satisfying \(|L(v)| \geq k\) 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 \((3, 1)^*\)-choosable.