On the Existence of Regular Sparse Anti-magic Squares of Odd Order

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 and d with d < n , an n × n array A based on { 0 , 1 , … , n d } is called a sparse anti-magic square of order n with density d , denoted by SAMS( n , d ), if each element of { 1 , 2 , … , n d } occurs exactly one entry of A , and its row-sums, column-sums and two main diagonal-sums constitute a set of 2 n + 2 consecutive integers. An SAMS( n , d ) is called regular if there are exactly d non-zero elements (or 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 odd order, and it is proved that for any odd n , there exists a regular SAMS( n , d ) if and only if 2 ≤ d ≤ n - 1 .