We present some results on two-path convexity in clone-free regular multipartite tournaments. After proving a structural result for regular multipartite tournaments with convexly independent sets of a given size, we determine tight upper bounds for their size (called the rank) in clone-free regular bipartite and tripartite tournaments. We use this to determine tight upper bounds for the Helly a...

The scheduling of social tournaments has attracted significant attention in recent years, as they arise in many practical applications and induce highly combinatorial problems. This research proposes a high-level modeling of social tournaments and a local search algorithm for finding solutions to the models. The effectiveness of the approach is demonstrated on three classes of applications: soc...

The chromatic number of the categorical product of two n-tournaments can be strictly smaller than n. We show that min{χ(S × T ) : S and T are n-tournaments} is asymptotically equal to λn, where 12 ≤ λ ≤ 2 3 .

In 4] it is conjectured that all diregular c-partite tournaments, with c 4, are pan-cyclic. In this paper we show that all diregular c-partite tournaments, with c 5, are in fact vertex-pancyclic.

Study Design: Descriptive epidemiology study. Methods: Injury data for all players (n 1⁄4 78) selected for 1 national Rugby Union team over a 3-year period were analyzed using the international consensus statement methods. Player height (cm) and mass (kg) were recorded. Tournaments were grouped for comparisons as: autumn tournaments (2012 and 2013), Rugby World Cup (RWC; 2011), Six Nations (201...

The tournament N5 can be obtained from the transitive tournament on {1, . . . , 5}, with the natural order, by reversing the edges between successive vertices. Tournaments that do not have N5 as a subtournament are said to omit N5. We describe the structure of tournaments that omit N5 and use this with Kruskal’s Tree Theorem to prove that this class of tournaments is well-quasi-ordered. The pro...

The Erdős-Hajnal Conjecture states that for every given H there exists a constant c(H) > 0 such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size at least |V (G)|c(H). The conjecture is still open. However some time ago its directed version was proved to be equivalent to the original one. In the directed version graphs are replaced by to...

Tournaments are widely used in organizations, explicitly or implicitly, to reward the best-performing employees, e.g., through promotion or bonuses, and/or to punish the worst-performing employees, e.g., through firing or unfavorable job assignments. We explore the impact of the allocation of prizes on the effectiveness of tournament incentive schemes. We show that while multiple prize allocati...

A tournament is acyclically indecomposable if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The profile of a tournament T is the function φT which counts for each integer n the n...

We extend Huber’s (1963) inequality for the joint distribution function of negative dependent scores in round-robin tournaments. As a byproduct, this extension implies convergence probability maximal score tournaments more general setting.