Constructions of regular sparse anti-magic squares

TL;DR

This paper studies the existence of regular sparse anti-magic squares (SAMS) with specific properties, proving their existence for certain orders and densities based on modular conditions.

Contribution

It establishes necessary and sufficient conditions for the existence of regular SAMS(n,d) when n ≡ 1,5 mod 6, filling a gap in graph labeling and combinatorial design literature.

Findings

Regular SAMS(n,d) exist for all n ≡ 1,5 mod 6 with 2 ≤ d ≤ n-1.

The paper provides constructive proofs for the existence of these squares.

It links the existence of SAMS to properties of graph labelings and combinatorial arrangements.

Abstract

Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$ array $A$ based on $\{0,1,\cdots,nd\}$ is called \emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each element of $\{1,2,\cdots,nd\}$ occurs exactly one entry of $A$, and its row-sums, column-sums and two main diagonal sums constitute a set of $2n+2$ consecutive integers. An SAMS$(n,d)$ is called \emph{regular} if there are exactly $d$ positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $n\equiv1,5\pmod 6$, and it is proved that for any $n\equiv1,5\pmod 6$, there exists a regular SAMS$(n,d)$ if and only if $2\leq d\leq n-1$.