Cubic vertex-transitive non-Cayley graphs of order 12p

TL;DR

This paper classifies all cubic vertex-transitive non-Cayley graphs of order 12p, where p is prime, revealing 11 sporadic cases and one infinite family depending on p's congruence modulo 4.

Contribution

It provides a complete classification of these graphs, identifying all sporadic and infinite cases based on prime p and its properties.

Findings

11 sporadic graphs for p=5,7,17

One infinite family exists if p ≡ 1 (mod 4)

Unique graph for each order in the infinite family

Abstract

A graph is said to be {\em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order $12p$, where $p$ is a prime, is given. As a result, there are $11$ sporadic and one infinite family of such graphs, of which the sporadic ones occur when $p=5$, $7$ or $17$, and the infinite family exists if and only if $p\equiv1\ (\mod 4)$, and in this family there is a unique graph for a given order.