Improper colorability of planar graphs without prescribed short cycles

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 subgraph G [ V i ] induced by V i has maximum degree at most d i for i = 1 , 2 , … , k . It is known that planar graphs without cycles of length 4 or l for any l ∈ { 5 , 6 } are ( 1 , 1 , 0 ) -colorable. In this paper, we prove that planar graphs without cycles of length 4 or l for any l ∈ { 7 , 8 } are also ( 1 , 1 , 0 ) -colorable. Some conjectures and problems for further study are presented.