Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs
Loading...
Date
2024
Authors
Morris, Dave W.
Morris, Joy
Journal Title
Journal ISSN
Volume Title
Publisher
Centre for Discrete Mathematics and Computing
Abstract
A finite group G is said to be a non-DCI group if there exist subsets S1
and S2 of G, such that the associated Cayley digraphs C−→ay(G; S1) and
C−→ay(G; S2) are isomorphic, but no automorphism of G carries S1 to S2.
Furthermore, G is said to be a non-CI group if the subsets S1 and S2
can be chosen to be closed under inverses, so we have undirected Cayley
graphs Cay(G; S1) and Cay(G; S2).
We show that if p is a prime number, and the elementary abelian p-
group (Zp)r is a non-DCI group, then (Zp)r+3 is a non-CI group. In most
cases, we can also show that (Zp)r+2 is a non-CI group. In particular,
from Pablo Spiga’s proof that (Z3)8 is a non-DCI group, we conclude that
(Z3)10 is a non-CI group. This is the first example of a non-CI elementary abelian 3-group.
Description
Open access article. Creative Commons Attribution 4.0 International license (CC BY 4.0) applies
Keywords
Caley graphs , Elementary abelian groups , CI graphs , CI groups , Isomorphism
Citation
Morris, D. W., & Morris, J. (2024). Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs. Australasian Journal of Combinatorics, 90(1), 46-59.