On Fractional Realizations of Tournament Score Sequences

Contrary to popular belief, we can’t all be winners. Suppose 6 people compete in a chess tournament in which all pairs of players compete directly and no ties are allowed; i.e., 6 people compete in a ‘round robin tournament’. Each player is assigned a ‘score’, namely the number of games they won, and the ‘score sequence’ of the tournament is a list of the players’ scores. Determining whether a given potential score sequence actually is a score sequence proves to be difficult. For instance, (0, 0, 3, 3, 3, 6) is not feasible because two players cannot both have score 0. Neither is the sequence (1, 1, 1, 4, 4, 4) because the sum of the scores is 16, but only 15 games are played among 6 players. This so called ‘tournament score sequence problem’ (TSSP) was solved in 1953 by the mathematical sociologist H. G. Landau. His work inspired the investigation of round robin tournaments as directed graphs. We study a modification in which the TSSP is cast as a system of inequalities whose solutions form a polytope η-dimensional space. This relaxation allows us to investigate the possibility of fractional scores. If, in a ‘round-robin’-ish tournament, Players A and B play each other 3 times, and Player A wins 2 of the 3 games, we can record this interaction as a 2/3 score for Player A and a 1/3 score for Player B. This generalization greatly impacts the nature of possible score sequences. We will also entertain an interpretation of these fractional scores as probabilities predicting the outcome of a true round robin tournament. The intersection of digraph theory, polyhedral combinatorics, and linear programming is a relatively new branch of graph theory. These results pioneer research in this field

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem