Decomposition of planar graphs with forbidden configurations

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