(1,1,0)-coloring of planar graphs without cycles of length 4 and 6

A graph G is called improperly (d1,d2,…,dk)-colorable, or simply (d1,d2,…,dk)-colorable, if the vertex set of G can be partitioned into subsets V1,V2,…,Vk such that the graph G[Vi] induced by Vi has maximum degree at most di for 1≤i≤k. In 1976, Steinberg raised the following conjecture: every planar graph without 4- and 5-cycles is (0,0,0)-colorable. Up to now, this challenge conjecture is still open. In this paper, we prove that every planar graph without cycles of length 4 and 6 is (1,1,0)-colorable.