# Sparse Anti-magic Squares and Vertex-magic Labelings of Bipartite Graphs

MacDougall, James A. and Gray, I. D. (2006) Sparse Anti-magic Squares and Vertex-magic Labelings of Bipartite Graphs. Discrete Mathematics, 306 . pp. 2878-2892.

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A sparse anti-magic square is an $n \times n$ array whose non-zero entries are the consecutive integers $1, · · · ,m$ for some $m \leq n^2$ and whose row-sums and column-sums form a set of consecutive integers. We derive some basic properties of these arrays and provide constructions for several infinite families of them. Our main interest in these arrays is their application to constructing vertex-magic labelings for bipartite graphs.