On a conjecture concerning vertex-transitive graphs

TL;DR

This paper introduces a new, stronger minor relation for finite graphs and formulates a conjecture linking it to the Lovasz conjecture, offering potential insights into the symmetry properties of vertex-transitive graphs.

Contribution

It proposes a novel minor relation and a related conjecture that could imply the Lovasz conjecture and advance understanding of vertex-transitive graph symmetry.

Findings

Defined a stronger minor relation than the classical one.

Formulated a conjecture connecting the new relation to the Lovasz conjecture.

Provides a new approach to studying symmetry in vertex-transitive graphs.

Abstract

In this article we define a minor relation, which is stronger than the classical one, but too strong to become a well-quasi-order on the class of finite graphs. Nevertheless, with this terminology we are able to introduce a conjecture, which would imply the Lovasz conjecture and give an interesting insight on the symmetry of vertex-transitive graphs, if true. Though it could become an approach to solve the Lovasz conjecture. These ideas were first introduced by the author in his Diploma Thesis.