In this article, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays H(m, n; s, k) whenever the necessary conditions hold, that is, 3 ⩽ s ⩽n, 3 ⩽ k ⩽m, ms = nk and nk ≡ 0, 3 (mod 4). By constructing integer Heffter array sets, we prove the conjecture in the affirmative whenever k ⩾ s gcd( s, k ) is odd and s ≠ 3, 5, 6, 10.
Pellegrini, M. A., Traetta, T., Toward a Solution of Archdeacon's Conjecture on Integer Heffter Arrays, <<JOURNAL OF COMBINATORIAL DESIGNS>>, 2025; (N/A): N/A-N/A. [doi:10.1002/jcd.21983] [https://hdl.handle.net/10807/312216]
Toward a Solution of Archdeacon's Conjecture on Integer Heffter Arrays
Pellegrini, Marco Antonio
;
2025
Abstract
In this article, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays H(m, n; s, k) whenever the necessary conditions hold, that is, 3 ⩽ s ⩽n, 3 ⩽ k ⩽m, ms = nk and nk ≡ 0, 3 (mod 4). By constructing integer Heffter array sets, we prove the conjecture in the affirmative whenever k ⩾ s gcd( s, k ) is odd and s ≠ 3, 5, 6, 10.
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