Graphs with two crossings are 5-choosable

TL;DR

This paper extends Thomassen's theorem by proving that graphs with up to two crossings are also 5-choosable, broadening the class of graphs known to be list-colorable with five colors.

Contribution

The paper proves that graphs with at most two crossings are 5-choosable, expanding the understanding of list-colorability beyond planar graphs.

Findings

Graphs with at most two crossings are 5-choosable.

Extension of Thomassen's theorem to non-planar graphs with limited crossings.

Broader class of graphs proven to be 5-choosable.

Abstract

A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. Thomassen's theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with at most two crossings is 5-choosable.