Thomassen's Choosability Argument Revisited

TL;DR

This paper presents a new proof that all $K_5$-minor-free graphs are 5-choosable, avoiding Wagner's structure theorem, and discusses implications for the Hadwiger Conjecture.

Contribution

It provides a novel proof technique for 5-choosability of $K_5$-minor-free graphs without relying on Wagner's theorem, offering insights for the Hadwiger Conjecture.

Findings

All $K_5$-minor-free graphs are 5-choosable.

New proof avoids Wagner's structure theorem.

Potential approach for Hadwiger Conjecture.

Abstract

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.