Two Arc-Disjoint Hamiltonian Paths in Finite Two-Generated Abelian Cayley Digraphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917536643153920 |
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| author | Park, SangHyun |
| author_facet | Park, SangHyun |
| contents | We prove the finite abelian two-generator conjecture of Darijani--Miraftab--Witte Morris: every directed Cayley digraph on a finite abelian group with two distinct nonzero generators has two arc-disjoint Hamiltonian paths. The proof uses a cut-reflection theorem for Hamiltonian cut values in the family Cay(Z_k; a, a+1): if Z is the set of such values and N=k-1, then, with N-Z={N-z : z in Z}, dist(Z,N-Z)<=1. The proof uses sector-filling inequalities for primitive-ray multiplicities and an extremal graph recording pairs at minimal reflected distance. The estimate is sharp modulo parity: exact reflection occurs for odd k, while distance one occurs for even k. The second remaining cyclic family, Cay(Z_k; -a, a+1), is treated by an explicit quotient--fiber construction. We also prove the remaining three-factor case for Cartesian products of directed cycles. Together with the two-factor and at-least-four-factor theorems of Darijani--Miraftab--Witte Morris, this resolves their directed-cycle product conjecture for all numbers of factors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_27241 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two Arc-Disjoint Hamiltonian Paths in Finite Two-Generated Abelian Cayley Digraphs Park, SangHyun Combinatorics Group Theory We prove the finite abelian two-generator conjecture of Darijani--Miraftab--Witte Morris: every directed Cayley digraph on a finite abelian group with two distinct nonzero generators has two arc-disjoint Hamiltonian paths. The proof uses a cut-reflection theorem for Hamiltonian cut values in the family Cay(Z_k; a, a+1): if Z is the set of such values and N=k-1, then, with N-Z={N-z : z in Z}, dist(Z,N-Z)<=1. The proof uses sector-filling inequalities for primitive-ray multiplicities and an extremal graph recording pairs at minimal reflected distance. The estimate is sharp modulo parity: exact reflection occurs for odd k, while distance one occurs for even k. The second remaining cyclic family, Cay(Z_k; -a, a+1), is treated by an explicit quotient--fiber construction. We also prove the remaining three-factor case for Cartesian products of directed cycles. Together with the two-factor and at-least-four-factor theorems of Darijani--Miraftab--Witte Morris, this resolves their directed-cycle product conjecture for all numbers of factors. |
| title | Two Arc-Disjoint Hamiltonian Paths in Finite Two-Generated Abelian Cayley Digraphs |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2605.27241 |