Decomposition of planar graphs with forbidden configurations

A $(d,h)$-decomposition of a graph $G$ is an ordered pair $(D, H)$ such that $H$ is a subgraph of $G$ of maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ with maximum out-degree at most $d$. In this paper, we prove that for $l \in \{5, 6, 7, 8, 9\}$, every planar graph without $4$- and $l$-cycles is $(2,1)$-decomposable. As a consequence, for every planar graph $G$ without $4$- and $l$-cycles, there exists a matching $M$, such that $G - M$ is $3$-DP-colorable and has Alon-Tarsi number at most $3$. In particular, $G$ is $1$-defective $3$-DP-colorable, $1$-defective $3$-paintable and 1-defective 3-choosable. These strengthen the results in [Discrete Appl. Math. 157~(2) (2009) 433--436] and [Discrete Math. 343 (2020) 111797].