$(1,0,0)$-Colorability of planar graphs without prescribed short cycles

Abstract

Let $d_1, d_2,\ldots ,d_k$ be $k$ non-negative integers. A graph $G$ is $(d_1,d_2,\ldots ,d_k)$-colorable, if the vertex set of $G$ can be partitioned into subsets $V_1,V_2,\ldots ,V_k$ such that the subgraph $G[V_i]$ induced by $V_i$ has maximum degree at most $d_i$ for $i=1,2,\ldots ,k$. Let $\digamma$ be the family of planar graphs with cycles of length neither 4 nor 8. In this paper, we prove that a planar graph in $\digamma$ is $(1,0,0)$-colorable if it has no cycle of length $k$ for some $k\in \{7,9\}$. 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 $4<i<j\le 9$.