# Cubic vertex-transitive graphs of girth seven

## TL;DR

This paper classifies all cubic vertex-transitive graphs with girth 7, identifying their structures and conditions for arc-transitivity, including specific families and exceptional graphs.

## Contribution

It provides a complete classification of cubic vertex-transitive graphs of girth 7 based on their signature and structural properties.

## Key findings

- Graphs are either truncations, skeletons of rotary maps, Cayley graphs, or specific Petersen and Coxeter graphs.
- If all edges are in the same number of 7-cycles, the graph is arc-transitive.
- The classification includes explicit families and exceptional cases.

## Abstract

In this paper we classify cubic vertex-transitive graphs of girth $7$, based on their signature. Such a graph is either a truncation of an arc-transitive dihedral scheme on a $7$-regular graph, the skeleton of a rotary map of type $\{7,3\}$, a member of an infinite family of Cayley graphs, or is one of the of the generalised Petersen graphs $\text{Pet}(13,5)$, $\text{Pet}(15,4)$, $\text{Pet}(17,4)$ or the Coxeter graph. We show that for a cubic vertex-transitive graphs $\Gamma$ of girth $7$, if every edge of $\Gamma$ is contained in the same number of $7$-cycles, then $\Gamma$ is also arc-transitive.