Every planar graph with girth at least 5 is (1,9)-colorable

Abstract

A graph G is (d1,…,dk)-colorable if the vertex set V(G) can be partitioned into k sets V1,…,Vk such that the subgraph G[Vi] has maximum degree at most di for 1≤i≤k. Montassier and Ochem [11], Choi and Raspaud [5], independently, posed an open question whether every planar graph of girth at least 5 is (1,4)-colorable. In this paper, we prove that every planar graph of girth at least 5 is (1,9)-colorable, which improves the result of Choi, Choi, Jeong and Suh who showed that every planar graph of girth at least 5 is (1,10)-colorable [J. Graph Theory 84 (2017) 521–535].