Planar graphs without cycles of length 4 or 5 are (3,0,0)-colorable

We study Steinberg's Conjecture. A graph is (c1,c2,…,ck)-colorable if the vertex set can be partitioned into k sets V1,V2,…,Vk such that for every i with 1≤i≤k the subgraph G[Vi] has maximum degree at most ci. We show that every planar graph without 4- or 5-cycles is (3,0,0)-colorable. This is a relaxation of Steinberg's Conjecture that every planar graph without 4- or 5-cycles is properly 3-colorable (i.e.,  (0,0,0)-colorable).