The Alon-Tarsi number of planar graphs without cycles of lengths $4$ and $l$

TL;DR

This paper investigates the Alon-Tarsi number of specific planar graphs lacking 4-cycles and certain l-cycles, establishing bounds and implications for their paintability and defective colorability.

Contribution

It proves the existence of a matching reducing the Alon-Tarsi number to 3 in such graphs, extending understanding of their coloring properties.

Findings

Existence of a matching M with AT(G-M) ≤ 3

Planar graphs without 4-cycles and l-cycles are 1-defective 3-paintable

New bounds on Alon-Tarsi numbers for these graphs

Abstract

This paper proves that if $G$ is a planar graph without 4-cycles and $l$-cycles for some $l\in\{5, 6, 7\}$, then there exists a matching $M$ such that $AT(G-M)\leq 3$. This implies that every planar graph without 4-cycles and $l$-cycles for some $l\in\{5, 6, 7\}$ is 1-defective 3-paintable.