Sparse Semi-Magic Squares and Magic Labelings of Directed Graphs

This paper will introduce the concept of vertex-magic edge labelings (VMEL) for digraphs. A digraph has a vertex-magic edge labeling if we can label the edges of the digraph such that the sum of the edges going in to any vertex equals the sum going out at any vertex and that sum is constant across all vertices in the digraph. Many examples will be given. The connections to sparse semi-magic squares are also given. Then, new sparse semi-magic squares are constructed that lead to both VMEL and magic labelings for various classes of digraphs.