Decomposition of planar graphs with forbidden configurations

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) of maximum out-degree at most d. In this paper, we prove that for l ∈ {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].