On Color Preserving Automorphisms of Cayley Graphs of Odd Square-free Order

TL;DR

This paper investigates the structure of color-preserving automorphisms in Cayley graphs of groups with odd square-free order, identifying unique non-CCA graphs and their relation to group decompositions.

Contribution

It proves the uniqueness of the non-CCA Cayley graph of the group of order 21 and characterizes non-CCA graphs of odd square-free order groups as products involving this graph.

Findings

Unique non-CCA Cayley graph of order 21 identified

Non-CCA Cayley graphs of odd square-free order groups are products involving the order 21 graph

Structure of automorphisms in Cayley graphs of specific group orders clarified

Abstract

An automorphism $\alpha$ of a Cayley graph $Cay(G,S)$ of a group $G$ with connection set $S$ is color-preserving if $\alpha(g,gs) = (h,hs)$ or $(h,hs^{-1})$ for every edge $(g,gs)\in E(Cay(G,S))$. If every color-preserving automorphism of $Cay(G,S)$ is also affine, then $Cay(G,S)$ is a CCA (Cayley color automorphism) graph. If every Cayley graph $Cay(G,S)$ is a CCA graph, then $G$ is a CCA group. Hujdurovi\'c, Kutnar, D.W. Morris, and J. Morris have shown that every non-CCA group $G$ contains a section isomorphic to the nonabelian group $F_{21}$ of order $21$. We first show that there is a unique non-CCA Cayley graph $\Gamma$ of $F_{21}$. We then show that if $Cay(G,S)$ is a non-CCA graph of a group $G$ of odd square-free order, then $G = H\times F_{21}$ for some CCA group $H$, and $Cay(G,S) = Cay(G,T)\Box\Gamma$.