Decomposing a planar graph without triangular 4-cycles into a matching and a 3-colorable graph

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]. In this paper, we prove that every planar graph without triangular 4-cycles is (1,1,1)-colorable. Consequently, such planar graphs can be decomposed into a matching and a 3-colorable graph.