(1,0,0)-colorability of planar graphs without cycles of length 4, 5 or 9

Let d1,d2,…,dk be k nonnegative integers. A graph G=(V,E) is improperly (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. Let Ϝ denote the family of planar graphs with cycles of length neither 4 nor 5. A challenging conjecture proposed by Steinberg asserts that every one in Ϝ is (0,0,0)-colorable. Motivated by the conjecture, a few authors studied the improper colorability of Ϝ. Along the thread of improper colorability, it is known that every one in Ϝ is (3,0,0)- and (1,1,0)-colorable. In this paper, we show that a member in Ϝ is (1,0,0)-colorable if it has no cycle of length 9.