Arc-disjoint Hamiltonian paths in Cayley digraphs with a normal difference subgroup
Abstract
Let G=⟨ a,b⟩ be a finite group with distinct nonidentity generators, and suppose that H=⟨ ba−1⟩ is normal. We determine exactly which complementary generator choices on the nonterminal cosets of H extend to two arc-disjoint Hamiltonian paths with both terminal vertices in H. The associated return residue must avoid an explicit set of at most six residues. When [G:H]≥3, any two nonterminal cosets may be left free while all other choices are prescribed; two is best possible. Every such Cayley digraph has two arc-disjoint Hamiltonian paths; this includes all two-generator Cayley digraphs of finite abelian groups. The criterion follows from a complete distance formula for the Hamiltonian cut spectra of Cay(ℤq;{D,D+1}). Their unique nondegenerate exception comes from the odd central coefficient (q−1)/2. The geometric ingredient is a confinement theorem: two sector-filled integer sequences satisfying a length inequality, with overlapping ranges and distance at least two, have one sequence contained in a single gap of the other. Sector filling bounds alternation, and the length inequality excludes interlacing. A self-contained proof of the primitive-ray cycle profile follows the intersection geometry of Curran and Witte.
Related
BibTeX
@misc{park2026arcdisjoint,
title = {{Arc-disjoint Hamiltonian paths in Cayley digraphs with a normal difference subgroup}},
author = {SangHyun Park},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03763},
eprint = {2610.03763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03763},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Arc-disjoint Hamiltonian paths in Cayley digraphs with a normal difference subgroup
AU - SangHyun Park
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03763
UR - https://arxiv.org/abs/2610.03763
AB - Let G=⟨ a,b⟩ be a finite group with distinct nonidentity generators, and suppose that H=⟨ ba^−1⟩ is normal. We determine exactly which complementary generator choices on the nonterminal cosets of H extend to two arc-disjoint Hamiltonian paths with both terminal vertices in H. The associated return residue must avoid an explicit set of at most six residues. When [G:H]≥3, any two nonterminal cosets may be left free while all other choices are prescribed; two is best possible. Every such Cayley digraph has two arc-disjoint Hamiltonian paths; this includes all two-generator Cayley digraphs of finite abelian groups. The criterion follows from a complete distance formula for the Hamiltonian cut spectra of Cay(ℤ_q;{D,D+1}). Their unique nondegenerate exception comes from the odd central coefficient (q−1)/2. The geometric ingredient is a confinement theorem: two sector-filled integer sequences satisfying a length inequality, with overlapping ranges and distance at least two, have one sequence contained in a single gap of the other. Sector filling bounds alternation, and the length inequality excludes interlacing. A self-contained proof of the primitive-ray cycle profile follows the intersection geometry of Curran and Witte.
ER -