Further constructions of square integer relative Heffter arrays

TL;DR

The paper resolves open existence problems for square integer relative Heffter arrays with parameter $k=3$ by constructing two new infinite families with $t=6$ that are strippable and applicable when $n \equiv 0$ or $3 \pmod{4}$; combined with known results for $t \in \{1,2,n,2n\}$ (when $n \equiv 1 \pmod{4}$) and $t \in \{3, n, 2n\}$ (when $n \equiv 3 \pmod{4}$), these constructions yield a complete existence criterion for prime $n$. The authors employ cyclically diagonal layouts to build $IH_6(n;3)$ arrays with explicit support and a primary transversal, ensuring strippability, and they derive necessary conditions on $t$ and $n$ to narrow the search. They also analyze small-$t$ scenarios (t=4,5) and show partial results, including a nonexistence result for strippable $IH_4(4;3)$ and existence of $IH_5(5;3)$. Overall, the work closes several gaps in the existence landscape for square integer relative Heffter arrays with $k=3$ in the prime-$n$ regime, enabling applications in related combinatorial designs.

Abstract

A square integer relative Heffter array is an $n \times n$ array whose rows and columns sum to zero, each row and each column has exactly $k$ entries and either $x$ or $-x$ appears in the array for every $x \in \mathbb{Z}_{2nk+t}\setminus J$, where $J$ is a subgroup of size $t$. There are many open problems regarding the existence of these arrays. In this paper we construct two new infinite families of these arrays with the additional property that they are strippable. These constructions complete the existence theory for square integer relative Heffter arrays in the case where $k=3$ and $n$ is prime.

Figures (7)

• Figure 1: General construction of an $IH_t(n;3)$ for $n=11$
• Figure 2: General construction of an $IH_t(n;k)$ for $n=12$
• Figure : Known existence of an $IH_t(n;k)$
• Figure : Row entries for an $IH_6(n;3)$ for $n \equiv 3 \pmod{4}$
• Figure : Column entries for an $IH_6(n;3)$ for $n \equiv 3 \pmod{4}$

Theorems & Definitions (23)

• Example 1.1
• Definition 1.2
• Example 1.3
• Theorem 1.4
• Corollary 1.5
• proof
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2