Orienting and separating distance-transitive graphs

TL;DR

This paper identifies seven specific cubic distance-transitive graphs with unique ultrahomogeneous properties related to oriented cycles, and introduces a method to construct related Cayley digraphs with minimal cycle separation, initiating a broader study.

Contribution

It reveals a special ultrahomogeneous property in seven cubic distance-transitive graphs and proposes a new construction of related Cayley digraphs with minimal cycle separation.

Findings

Seven cubic distance-transitive graphs have unique ultrahomogeneous properties.

A new construction method for related Cayley digraphs with minimal cycle separation.

Initiation of a broader study on ultrahomogeneous properties in distance-transitive graphs.

Abstract

It is shown that exactly 7 distance-transitive cubic graphs among the existing 12 possess a particular ultrahomogeneous property with respect to oriented cycles realizing the girth that allows the construction of a related Cayley digraph with similar ultrahomogeneous properties in which those oriented cycles appear minimally "pulled apart", or "separated" and whose description is truly beautiful and insightful. This work is proposed as the initiation of a study of similar ultrahomogeneous properties for distance-transitive graphs in general with the aim of generalizing to constructions of similar related "separator" Cayley digraphs.