PreprintCC BY 4.0arXiv
Isomorphisms of abelian Cayley graphs with their natural edge-colouring
Abstract
We prove that if φ is an isomorphism between two connected Cayley graphs of abelian groups, and φ respects the natural edge-colourings of the Cayley graphs, then φ is the composition of a group isomorphism and a colour-preserving graph automorphism. This implies that if every colour-preserving automorphism of a connected abelian Cayley graph Cay(G;S) is an affine map, then the same is true for every colour-permuting automorphism. We also show that this property holds if and only if the subgroup generated by { s ∈ S | 2s ≠ c } ∪ {c} has index ≤ 2 for every element c of order $2$ in G.
Related
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BibTeX
@misc{alimirzaei2026isomorphisms,
title = {{Isomorphisms of abelian Cayley graphs with their natural edge-colouring}},
author = {Shirin Alimirzaei and Dave Witte Morris},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09563},
eprint = {2609.09563},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09563},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Isomorphisms of abelian Cayley graphs with their natural edge-colouring
AU - Shirin Alimirzaei
AU - Dave Witte Morris
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09563
UR - https://arxiv.org/abs/2609.09563
AB - We prove that if φ is an isomorphism between two connected Cayley graphs of abelian groups, and φ respects the natural edge-colourings of the Cayley graphs, then φ is the composition of a group isomorphism and a colour-preserving graph automorphism. This implies that if every colour-preserving automorphism of a connected abelian Cayley graph Cay(G;S) is an affine map, then the same is true for every colour-permuting automorphism. We also show that this property holds if and only if the subgroup generated by { s ∈ S | 2s ≠ c } ∪ {c} has index ≤ 2 for every element c of order $2$ in G.
ER -