Regular sparse anti-magic squares with the second maximum density

Abstract

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0,1,⋯,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 form a set of 2n+2 consecutive integers. An SAMS(n,d) is called regular if there are 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 with the second maximum density, i.e., d=n−2, and it is shown that there exists a regular SAMS(n,n−2) if and only if n≥4.