Decomposition of planar graphs with forbidden configurations

TL;DR

This paper proves that certain planar graphs without specific cycles can be decomposed into subgraphs with bounded degree and acyclic orientations, leading to new coloring and choosability results.

Contribution

It establishes $(2,1)$-decomposability for planar graphs without 4- and l-cycles for l in {5,6,7,8,9}, improving understanding of their colorability and structure.

Findings

Planar graphs without 4- and l-cycles are $(2,1)$-decomposable.

Such graphs are 3-DP-colorable after removing a matching.

These graphs are 1-defective 3-DP-colorable, 3-paintable, and 3-choosable.

Abstract

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].