The existence spectrum for regular sparse anti-magic squares

Abstract

Magic squares and anti-magic squares are the important research objects in combinatorics, and they are also well used in constructing graph labelings. Suppose that n and d are two positive integers with d<n. An n×n array A based on X={0,1,2,⋯,nd} is called a sparse anti-magic square of order n with density d, denoted by SAMS(n,d), if every non-zero element of X occurs exactly once in A and the set of the row-sums, column-sums and two main diagonal-sums of A consists of 2n+2 consecutive integers. An SAMS(n,d) is called regular if there are exactly d non-zero elements in each row, each column and each main diagonal, denoted by RSAMS(n,d). In this paper, the existence of an RSAMS(n,d) is investigated and it is proved that a necessary condition for the existence of an RSAMS(n,d) is also sufficient, i.e., there exists an RSAMS(n,d) if and only if 2≤d<n.