Choosability with union separation of planar graphs without intersecting triangles

Abstract

Let G be a graph, and let $k,s$ be two positive integers. A k-list assignment of G is called a $(k,k+s)$-list assignment if, for any two adjacent vertices u and v, $\left|L\right(u)\cup L(v\left)\right|\ge k+s$. A graph G is said to be $(k,k+s)$-choosable if, for every given $(k,k+s)$-list assignment, it always admits a proper coloring π such that $\pi \left(v\right)\in L\left(v\right)$ for every $v\in V\left(G\right)$. In this paper, we demonstrate that every planar graph without intersecting triangles is $(3,7)$-choosable. This strengthens a result which asserts that every triangle-free planar graph is $(3,7)$-choosable.