5-choosability of graphs with crossings far apart

TL;DR

This paper proves that graphs with crossings sufficiently far apart are 5-choosable, extending the known result for planar graphs and allowing some vertices with smaller lists under certain conditions.

Contribution

It provides a new proof for 5-choosability of planar graphs and extends this to graphs with crossings far apart, including vertices with smaller lists.

Findings

Graphs with crossings at least 15 apart are 5-choosable.

Some vertices can have list size four if they are far from crossings and each other.

The proof generalizes the 5-choosability of planar graphs to a broader class.

Abstract

We give a new proof of the fact that every planar graph is 5-choosable, and use it to show that every graph drawn in the plane so that the distance between every pair of crossings is at least 15 is 5-choosable. At the same time we may allow some vertices to have lists of size four only, as long as they are far apart and far from the crossings.