Completing the Existence Problem for Integer Relative Heffter Arrays Hk(n;k)
Abstract
Heffter arrays, introduced by Archdeacon in 2015, are combinatorial structures with applications to cyclic cycle systems and biembeddings of graphs on surfaces. Costa, Morini, Pasotti and Pellegrini proposed in 2020 the notion of relative Heffter arrays as a generalization of classical Heffter arrays, inspired by the concept of relative difference families. In their work, the existence problem of integer relative Heffter arrays Hk(n;k) was solved for all k≠ 5, while the case k=5 and n ≡ 0 pmod4 remained open, apart from two sporadic examples with n=8,16. In this article, we consider this open problem and we construct an H5(n;5) for every n≡ 0 pmod4, n≥ 12. As a consequence, the existence problem of integer relative Hk(n;k) is completely settled.
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BibTeX
@misc{mella2026completing,
title = {{Completing the Existence Problem for Integer Relative Heffter Arrays $\mathrmH_k(n;k)$}},
author = {Lorenzo Mella},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24399},
eprint = {2609.24399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24399},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Completing the Existence Problem for Integer Relative Heffter Arrays H_k(n;k)
AU - Lorenzo Mella
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24399
UR - https://arxiv.org/abs/2609.24399
AB - Heffter arrays, introduced by Archdeacon in 2015, are combinatorial structures with applications to cyclic cycle systems and biembeddings of graphs on surfaces. Costa, Morini, Pasotti and Pellegrini proposed in 2020 the notion of relative Heffter arrays as a generalization of classical Heffter arrays, inspired by the concept of relative difference families. In their work, the existence problem of integer relative Heffter arrays H_k(n;k) was solved for all k≠ 5, while the case k=5 and n ≡ 0 pmod4 remained open, apart from two sporadic examples with n=8,16. In this article, we consider this open problem and we construct an H_5(n;5) for every n≡ 0 pmod4, n≥ 12. As a consequence, the existence problem of integer relative H_k(n;k) is completely settled.
ER -