On subgroup perfect codes and regular sets in generalized Cayley graphs
Abstract
A subset C of the vertex set of a graph Γ is said to be an (a,b)-regular set if C induces an a-regular subgraph and every vertex outside C is adjacent to exactly b vertices in C. A (0,1)-regular set is called a perfect code. Let G be a finite group and let α∈Aut(G) be such that α2=id. Let ωα(G)={α(g−1)g:g∈ G}. For a subset S⊆ G satisfying α(S)=S−1 and S∩ωα(G)=∅, the generalized Cayley graph of (G,α) with respect to S is the graph with vertex set G in which two distinct elements g,h∈ G are adjacent if and only if α(g−1)h∈ S. In this paper, we prove two main results. First, let H be an α-invariant subgroup of a finite group G, and let P be an α-invariant Sylow 2-subgroup of H. We show that H is a perfect code of (G,α) if and only if P[H,α] is a perfect code of (G,α), where [H,α]=⟨ h−1α(h):h∈ H⟩. Second, we consider finite groups G possessing a cyclic subgroup A of index 2 that is invariant under an involutory automorphism α. We show that ωα(G) is a subgroup of A. Using this result, we give a complete characterization of all pairs (a,b) for which an α-invariant subgroup H of G is an (a,b)-regular set of (G,α). In particular, our characterization applies to all finite groups possessing a characteristic cyclic subgroup of index 2, for every involutory automorphism α.
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BibTeX
@misc{khaefi2026subgroup,
title = {{On subgroup perfect codes and regular sets in generalized Cayley graphs}},
author = {Yasamin Khaefi and Fateme Sadat Seiedali},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00893},
eprint = {2611.00893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00893},
}
RIS
TY - UNPB
TI - On subgroup perfect codes and regular sets in generalized Cayley graphs
AU - Yasamin Khaefi
AU - Fateme Sadat Seiedali
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00893
UR - https://arxiv.org/abs/2611.00893
AB - A subset C of the vertex set of a graph Γ is said to be an (a,b)-regular set if C induces an a-regular subgraph and every vertex outside C is adjacent to exactly b vertices in C. A (0,1)-regular set is called a perfect code. Let G be a finite group and let α∈Aut(G) be such that α^2=id. Let ω_α(G)={α(g^−1)g:g∈ G}. For a subset S⊆ G satisfying α(S)=S^−1 and S∩ω_α(G)=∅, the generalized Cayley graph of (G,α) with respect to S is the graph with vertex set G in which two distinct elements g,h∈ G are adjacent if and only if α(g^−1)h∈ S. In this paper, we prove two main results. First, let H be an α-invariant subgroup of a finite group G, and let P be an α-invariant Sylow 2-subgroup of H. We show that H is a perfect code of (G,α) if and only if P[H,α] is a perfect code of (G,α), where [H,α]=⟨ h^−1α(h):h∈ H⟩. Second, we consider finite groups G possessing a cyclic subgroup A of index 2 that is invariant under an involutory automorphism α. We show that ω_α(G) is a subgroup of A. Using this result, we give a complete characterization of all pairs (a,b) for which an α-invariant subgroup H of G is an (a,b)-regular set of (G,α). In particular, our characterization applies to all finite groups possessing a characteristic cyclic subgroup of index 2, for every involutory automorphism α.
ER -