Handicap distance antimagic graphs and incomplete tournaments

Let G = (V,E) be a graph of order n. A bijection f : V → {1, 2, . . . , n} is called a distance magic labeling of G if there exists a positive integer μ such that ∑ u∈N(v) f(u) = μ for all v ∈ V, where N(v) is the open neighborhood of v. The constant μ is called the magic constant of the labeling f. Any graph which admits a distance magic labeling is called a distance magic graph. The bijection f : V → {1, 2, . . . , n} is called a d-distance antimagic labeling of G if for V = {v1, v2, . . . , vn} the sums ∑ u∈N(vi) f(u) form an arithmetic progression with difference d. We introduce a generalization of the well-known notion of magic rectangles called magic rectangle sets and use it to find a class of graphs with properties derived from the distance magic graphs. Then we use the graphs to construct a special kind of incomplete round robin tournaments, called handicap tournaments.