The Alon-Tarsi number of subgraphs of a planar graph

TL;DR

This paper explores the Alon-Tarsi number in planar graphs, constructing specific examples where removal of certain subgraphs prevents 3-choosability, but also proves that a forest can be removed to ensure 3-choosability.

Contribution

It introduces new constructions of planar graphs with specific coloring properties and proves a universal result about forests ensuring 3-choosability after removal.

Findings

Constructed planar graphs with non-3-choosable subgraphs

Proved existence of a forest F in any planar graph with Alon-Tarsi number ≤ 3 after removal

Demonstrated that G - E(F) is 3-paintable and 3-choosable

Abstract

This paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\Delta(H) \le 3$, $G_1-E(H)$ is not $3$-choosable, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G - E(F)$ is at most $3$, and hence $G - E(F)$ is 3-paintable and 3-choosable.