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

Let d 1 , d 2 , … , d k be k nonnegative integers. A graph G = ( V , E ) is improperly ( d 1 , d 2 , … , d k ) -colorable, if the vertex set V of G can be partitioned into subsets V 1 , V 2 , … , V k such that the subgraph G [ V i ] induced by V i has maximum degree at most d i 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.