Towards Characterization of 5-List-Colorability of Toroidal Graphs

TL;DR

This paper uses computer-assisted enumeration to identify minimal obstructions for 5-choosability in toroidal graphs with a specific triangle configuration, supporting the conjecture linking 5-choosability and 5-colorability.

Contribution

It characterizes a class of obstructions for 5-choosability in toroidal graphs and verifies the absence of additional obstructions with a particular property, advancing understanding of graph colorability.

Findings

No additional obstructions with the property were found.

Supports the conjecture that 5-choosability and 5-colorability are equivalent for toroidal graphs.

Provides a comprehensive list of minimal obstructions under the given conditions.

Abstract

Through computer-assisted enumeration, we list minimal obstructions for 5-choosability of graphs on the torus with the following additional property: There exists a cyclic system of non-contractible triangles around the torus where the consecutive triangles are at distance at most four. This condition is satisfied by all previously known obstructions, and we verify that there are no additional obstructions with this property. This supports the conjecture that a toroidal graph is 5-choosable if and only if it is 5-colorable.