Planar graphs with cycles of length neither 4 nor 7 are $(3,0,0)$-colorable

Abstract

Let $d_1,d_2,\dots ,d_k$ be $k$ non-negative integers. A graph $G$ is $(d_1,d_2,\dots ,d_k)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,V_2,\dots ,V_k$ such that the subgraph $G\left[V_i\right]$ induced by $V_i$ has maximum degree at most $d_i$ for $1\le i\le k$. It is known that planar graphs with cycles of length neither 4 nor $k$, $k\in \{5,6\}$, are $(3,0,0)$-colorable. In this paper, we show that planar graphs with cycles of length neither 4 nor 7 are also $(3,0,0)$-colorable.