Isomorphism factorizations of the complete graph into Cayley graphs on CI-groups

TL;DR

This paper characterizes when complete graphs can be decomposed into isomorphic Cayley graphs on CI-groups, providing necessary and sufficient conditions and a construction method for such factorizations.

Contribution

It establishes a complete characterization of CI-groups allowing edge-partitions of complete graphs into isomorphic Cayley graphs, and introduces a construction method for these factorizations.

Findings

Necessary and sufficient condition for CI-groups to admit such factorizations.

A construction method for isomorphic factorizations into Cayley graphs.

Extension of classical graph decomposition results to Cayley graphs on CI-groups.

Abstract

Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this paper, we investigate isomorphic factorizations of complete graphs into Cayley graphs on CI-groups. Let $\Gamma=Cay(G,S)$ denote the Cayley graph of finite group $G$. We obtain a necessary and sufficient condition on CI-group $G$ so that the complete graph on $|G|$ vertices can be edge-partitioned into $k$-copies of Cayley graph of the same CI-group $G$ each isomorphic to $Cay(G,S)$ for some inverse-closed subset $S\subset G\setminus\{1\}$. Further we give a construction of isomorphic factorizations of the complete graph into Cayley graphs on CI-group.