Improper colorability of planar graphs with cycles of length neither 4 nor 6

Let d 1 , d 2 ,..., d k be k non-negative integers. A graph G is ( d 1 , d 2 ,..., d k )-colorable, if the vertex set of G can be partitioned into subsets V 1 , V 2 ,..., V k such that the graph G [ V i ] induced by V i has maximum degree at most d i for em =1, 2,..., k . In this paper, we prove that every planar graph with cycles of length neither 4 nor 6 is (3, 0, 0)-and (1, 1, 0)-colorable.