What is and is not inside a Cayley graph?

TL;DR

This paper investigates which graphs can be embedded in Cayley graphs, showing a specific cubic graph of girth 5 cannot be a subgraph of any minimal Cayley graph, while certain generalized Petersen graphs can.

Contribution

It demonstrates the existence of a cubic girth 5 graph not embeddable in any minimal Cayley graph and proves that generalized Petersen graphs with coprime parameters are induced subgraphs of minimal Cayley graphs.

Findings

A cubic girth 5 graph is not a subgraph of any minimal Cayley graph.

Generalized Petersen graphs G(n,k) with gcd(n,k)=1 are induced subgraphs of minimal Cayley graphs.

Abstract

In this note we show that there is a cubic graph of girth $5$ that is not a subgraph of any minimal Cayley graph. On the other hand, we show that any Generalized Petersen Graph $G(n,k)$ with $\gcd(n,k)=1$ is an induced subgraph of a minimal Cayley graph. These results give insights into two comments of L\'aszl\'o Babai in [L. Babai, \emph{Automorphism groups, isomorphism, reconstruction}. Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2, 1994].