Hamilton decompositions of regular tournaments

We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each \eta>0 every regular tournament G of sufficiently large order n contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.Comment: 38 pages, 2 figures. Added section sketching how we can extend our main result. To appear in the Proceedings of the LM

Dynamic Bradley–Terry modelling of sports tournaments

Summary.  In the course of national sports tournaments, usually lasting several months, it is expected that the abilities of teams taking part in the tournament will change over time. A dynamic extension of the Bradley–Terry model for paired comparison data is introduced to model the outcomes of sporting contests, allowing for time varying abilities. It is assumed that teams’ home and away abilities depend on past results through exponentially weighted moving average processes. The model proposed is applied to sports data with and without tied contests, namely the 2009–2010 regular season of the National Basketball Association tournament and the 2008–2009 Italian Serie A football season

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio