Cores of Vertex Transitive Graphs

TL;DR

This paper investigates the structure of vertex transitive graphs, focusing on cores and conditions under which their vertices can be partitioned into subsets inducing cores, revealing specific cases where such partitions exist or do not.

Contribution

It establishes conditions for partitioning vertex transitive graphs into cores, particularly for normal Cayley graphs and graphs with cores half their size, advancing understanding of their structural properties.

Findings

Normal Cayley graphs admit such partitions.

Vertex transitive graphs with cores half their size do admit partitions.

Graphs with smaller cores generally do not admit such partitions.

Abstract

A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions.