Planar graphs without cycles of length 4 or 5 are (2,0,0)-colorable

Let d1,d2,…,dk be k nonnegative integers. A graph G=(V,E) is (d1,d2,…,dk)-colorable, if the vertex set V of G can be partitioned into subsets V1,V2,…,Vk such that the subgraph G[Vi] induced by Vi has maximum degree at most di for i=1,2,…,k. Steinberg conjectured that planar graphs without cycles of length 4 or 5 are (0,0,0)-colorable. Hill et al. showed that every planar graph without cycles of length 4 or 5 is (3,0,0)-colorable. In this paper, we show that planar graphs without cycles of length 4 or 5 are (2,0,0)-colorable. For further study in this direction, some problems and conjectures are presented.