No abstract

No abstract

We consider the notion of a signed magic array, which is an m × n rectangular array with the same number of filled cells s in each row and the same number of filled cells t in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero sum. We attempt to make progress toward a characterization of for which (m, n, s, t) there exists such an array. This characterization is complete in the case where n = s and in the case where n = m; we also characterize three-fourths of the cases where n = 2m.

Let e 1 ; e 2 ; y; e n be a sequence of nonnegative integers such that the first non-zero term is not one. Let P n i¼1 e i ¼ ðq À 1Þ=2; where q ¼ p n and p is an odd prime. We prove that the complete graph on q vertices can be decomposed into e 1 C p n -factors, e 2 C p nÀ1 -factors, y; and e n C p -factors. r 2004 Elsevier Inc. All rights reserved.

No abstract