Every planar graph without triangles adjacent to cycles of length 3 or 6 is [formula omitted]-colorable

Abstract

Let c1,c2,…,ck be non-negative integers. A graph G is (c1,c2,…,ck)-colorable if the vertex set of G can be partitioned into k sets V1,V2,…,Vk, such that G[Vi], the subgraph induced by Vi, has maximum degree at most ci for i∈[k]. Wang and Xu (2015) posed some open problems, one of which is that every planar graph without adjacent triangles is (1,1,1)-colorable. In this paper, we prove that every planar graph without triangles adjacent to cycles of length 3 or 6 is (1,1,1)-colorable.