(10,0,0)-Colorability of planar graphs without prescribed short cycles

Abstract

Let d1,d2,…,dk be k non-negative integers. A graph G is (d1,d2,…,dk)-colorable, if the vertex set 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 be the family of planar graphs with cycles of length neither 4 nor 8. In this paper, we prove that a planar graph in ϝ is (1,0,0)-colorable if it has no cycle of length k for some k∈{7,9}. Together with other known related results, this completes a neat conclusion on the (1,0,0)-colorability of planar graphs without prescribed short cycles, more precisely, for every triple (4,i,j), planar graphs without cycles of length 4, i or j are (1,0,0)-colorable whenever 4