The Alon-Tarsi number of $K_5$-minor-free graphs

TL;DR

This paper investigates the Alon-Tarsi number of $K_5$-minor-free graphs, establishing upper bounds and structural modifications that reduce the number, thus advancing understanding of graph coloring properties.

Contribution

It proves three theorems providing upper bounds on the Alon-Tarsi number for $K_5$-minor-free graphs and identifies specific substructures that lower this number.

Findings

Alon-Tarsi number of $K_5$-minor-free graphs is at most 5

Existence of a matching reducing the Alon-Tarsi number to at most 4

Existence of a forest reducing the Alon-Tarsi number to at most 3

Abstract

In this paper, we show the following three theorems. Let $G$ be a $K_5$-minor-free graph. Then Alon-Tarsi number of $G$ is at most $5$, there exists a matching $M$ of $G$ such that the Alon-Tarsi number of $G-M$ is at most $4$, and there exists a forest $F$ such that the Alon-Tarsi number of $G-E(F)$ is at most $3$.