Regular Sparse Anti-Magic Squares with Maximum Density

Abstract

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An $n\times 7$ array based on $\{0,1,\dots ,nd\}$ is called a sparse anti-magic square of order $n$ with density $d$ ($d<n$), denoted by SAMS$(n,d)$, if its row-sums, column-sums, and two main diagonal sums constitute a set of $2n+2$ consecutive integers. A SAMS$(n,d)$ is called regular if there are $d$ positive entries in each row, each column, and each main diagonal. In this paper, some constructions of regular sparse anti-magic squares are provided and it is shown that there exists a regular SAMS$(n,d-1)$ if and only if $n\ge 4$.