Every planar graph without 3-cycles adjacent to 4-cycles and without 6-cycles is (1, 1, 0)-colorable

Abstract

Let $c_{1},c_{2},\ldots ,c_{k}$ be k non-negative integers. A graph G is $(c_{1},c_{2},\ldots ,c_{k})$-colorable if the vertex set can be partitioned into k sets $V_{1},V_{2},\ldots ,V_{k}$ such that for every $i,1\le i\le k$, the subgraph $G[V_{i}]$ has maximum degree at most $c_{i}$. Steinberg (Ann Discret Math 55:211–248, 1993) conjectured that every planar graph without 4- and 5-cycles is 3-colorable. Xu and Wang (Sci Math 43:15–24, 2013) conjectured that every planar graph without 4- and 6-cycles is 3-colorable. In this paper, we prove that every planar graph without 3-cycles adjacent to 4-cycles and without 6-cycles is (1, 1, 0)-colorable, which improves the result of Xu and Wang (Sci Math 43:15–24, 2013), who proved that every planar graph without 4- and 6-cycles is (1, 1, 0)-colorable.