Regular sparse anti-magic squares with small odd densities

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n × n array based on { 0 , 1 , ? , n d } 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 2 n + 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, we investigate the existence of regular sparse anti-magic squares with densities d = 3 , 5 and it is proved that there exists a regular SAMS ( n , 3 ) if and only if n ? 4 and there exists a regular SAMS ( n , 5 ) if and only if n ? 6 .