[Paper Review] Signed magic rectangles with three filled cells in each column

Abdollah Khodkar, Chris Leach+1 more|arXiv (Cornell University)|Dec 18, 2019

Graph Labeling and Dimension Problems4 citations

TL;DR

This paper establishes the necessary and sufficient conditions for the existence of signed magic rectangles with exactly three filled cells per column, proving that an SMR(m,n;k,3) exists if and only if 3 ≤ m,k ≤ n and mk = 3n. The construction relies on partitioning integer sets into near-orthogonal triples and leveraging known results on magic rectangles and Heffter arrays to ensure zero-sum rows and columns with symmetric integer entries.

ABSTRACT

A {\em signed magic rectangle} $SMR(m,n;k, s)$ is an $m imes n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (mk-1)/2\}$ if $mk$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mk/2\}$ if $mk$ is even, such that precisely $k$ cells in every row and $s$ cells in every column are filled, every integer from set $X$ appears exactly once in the array and the sum of each row and of each column is zero. In this paper, we prove that a signed magic rectangle $SMR(m,n;k, 3)$ exists if and only if $3\leq m,k\leq n$ and $mk=3n$.

Motivation & Objective

• To determine the necessary and sufficient conditions for the existence of signed magic rectangles SMR(m,n;k,3) with exactly three filled cells per column.
• To extend the theory of signed magic rectangles beyond previously studied cases with two filled cells per column.
• To resolve an open problem regarding the existence of SMR(m,n;k,3) by characterizing all valid parameter combinations.
• To provide constructive methods using integer partitions and orthogonal decompositions to build such rectangles when conditions are satisfied.

Proposed method

• Leverages Lemma 7 to transform existing magic rectangles with odd row sums into signed magic rectangles by centralizing entries around zero.
• Applies Theorem 8 to construct magic rectangles MR(m,km;ks,s) under specific parity and size constraints, which are then converted into SMR(m,km;ks,s) via symmetric shifting.
• Uses Proposition 9 to construct SMR(m,km;3k,3) when m, k, s are odd and satisfy 3 ≤ s ≤ m.
• Employs partitioning techniques to decompose the integer set X into disjoint triples summing to zero, ensuring near-orthogonality with column partitions.
• Applies Theorem 13 to verify the existence of SMR(m,n;k,3) when a partition of X is near-orthogonal to the column partition C1,…,Cn.
• Constructs explicit examples using symmetric triple sets like {±a, ±b, ±c} and their signed counterparts to satisfy row and column sum constraints.

Experimental results

Research questions

• RQ1 Under what conditions does a signed magic rectangle SMR(m,n;k,3) exist with exactly three filled cells per column and k filled cells per row?
• RQ2 Can the existence of SMR(m,n;k,3) be characterized solely by the equation mk = 3n and the bounds 3 ≤ m,k ≤ n?
• RQ3 How can the construction of such rectangles be systematically achieved when the necessary conditions are met?
• RQ4 What role do integer partitions into zero-sum triples and their orthogonality to column partitions play in the existence proof?
• RQ5 Are there constructive methods to generate SMR(m,n;k,3) for all valid parameter combinations?

Key findings

• An SMR(m,n;k,3) exists if and only if 3 ≤ m,k ≤ n and mk = 3n.
• When mk = 3n and m,k ≥ 3, the existence is guaranteed through constructive partitioning of the integer set X into zero-sum triples.
• The construction is valid for both odd and even n, with separate treatments for cases where 3 divides k or m.
• Explicit examples are provided for small cases such as SMR(10,30;9,3), SMR(4,12;9,3), and SMR(30,30;5,3), confirming the theoretical conditions.
• The method relies on near-orthogonal partitions of X into triples that sum to zero and are compatible with column and row constraints.
• The proof unifies prior results on magic rectangles and Heffter arrays, extending them to the signed case with three filled cells per column.