Sparse Semi-Magic Squares and Vertex-magic Labelings
I. D. Gray, James A. MacDougall · Ars Combinatoria · 2006
We introduce a generalisation of the traditional magic square which proves useful in the construction of magic labelings of graphs. An order n sparse semi-magic square is an $n \times n$ array containing the entries $1, 2, . . . ,m$ (for some $m < n^2$) once each with the remainder of its entries 0, and its rows and columns have a constant sum $k$. We discover some of the basic properties of such arrays and provide constructions for squares of all orders $n \geq 3$. We also show how these arrays can be used to produce vertex-magic labelings for certain families of graphs.