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 (Formula presented.) 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,..., v_n} the sums (Formula presented.) 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.