2025/10/09 by Fiorenza Morini, Morini, Fiorenza, Marco Antonio Pellegrini +1
Engineering · Mathematics · #05B20 #05B30 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2510.08302
openalex publication_date 2025/10/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
An integer Heffter array H(m,n;s;k) is an m× n partially filled array whose entries are the elements of a subset Ω⊂ ℤ such that \Ω,-Ω\ is a partition of the set \1,2,…,2nk\ and such that the following conditions are satisfied: each row contains s filled cells, each column contains k filled cells, the elements in every row and column add up to 0. It was conjectured by Dan Archdeacon that an integer \H(m,n;s;k) exists if and only if ms=nk, 3≤ s ≤ n, 3≤ k≤ m and nk≡ 0,3\pmod 4. In this paper, we provide new constructions of these objects that allow us to prove the validity of Archdeacon's conjecture in each admissible case, except when k=3,5 and s\not ≡ 0\pmod 4 is such that gcd(s,k)=1.