Improper colorability of planar graphs without prescribed short cycles

Let d1,d2,…,dkd1,d2,…,dk be kk non-negative integers. A graph GG is (d1,d2,…,dk)(d1,d2,…,dk)-colorable, if the vertex set of GG can be partitioned into subsets V1,V2,…,VkV1,V2,…,Vk such that the subgraph G[Vi]G[Vi] induced by ViVi has maximum degree at most didi for i=1,2,…,ki=1,2,…,k. It is known that planar graphs without cycles of length 4 or ll for any l∈{5,6}l∈{5,6} are (1,1,0)(1,1,0)-colorable. In this paper, we prove that planar graphs without cycles of length 4 or ll for any l∈{7,8}l∈{7,8} are also (1,1,0)(1,1,0)-colorable. Some conjectures and problems for further study are presented.