Ohodnocení grafů a rozpisy

Abstract

We will always have tournaments, because people want to know, who is better in games or sports. We aproximate a tournament with a simple graph, where teams will be vertices and a match between two teams in the tournament will be represented by an edge. Every graph, which aproximates a tournament, has to meet certain properties. In this thesis we describe tournaments with a vertex graph labeling, which aproximate total strength of the opponents of each team predominantly in equalized incomplete tournaments.

We assign strength to every team (i.e.~by their placement in the previsious tournament) and count the weight of all opponents for each team as the sum of labels of vertices adjacent to the corresponding vertex.

In this thesis we create a new arithmetric labeling of graphs, which is a generalization of distance magic labeling, fair labeling and handicap labeling. We show that this labeling is equivalent to handicap labeling for regular graphs. Existence of this new labeling is settled for $r$-regular graphs, where r=1 or r=2. Further this thesis provides necessary conditions for the existence of arithmetic graph labeling.