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 \(\{0, 1, \ldots, nd\}\) 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 \(n \geq 4\).