Preface Tournaments constitute perhaps the most well-studied class of directed graphs. One of the reasons for the interest in the theory of tournaments is the monograph Topics on Tournaments [58] by Moon published in 1968, covering all results on tournaments known up to this time. In particular, three results deserve special mention: in 1934 Rédei [60] proved that every tournament has a directe...

Let $T$ be a non-trivial tournament. An arc is emph{$t$-pancyclic} in $T$, if it is contained in a cycle of length $ell$ for every $tleq ell leq |V(T)|$. Let $p^t(T)$ denote the number of $t$-pancyclic arcs in $T$ and $h^t(T)$ the maximum number of $t$-pancyclic arcs contained in the same Hamiltonian cycle of $T$. Moon ({em J. Combin. Inform. System Sci.}, {bf 19} (1994), 207-214) showed that $...

We study some problems pertaining to the tournament equilibrium set (τ for short). A tournament H is a τ-retentive tournament if there is a tournament T which has a minimal τ-retentive set R such that T [R] is isomorphic to H. We study τ-retentive tournaments and achieve many significant results. In particular, we prove that there are no τretentive tournaments of size 4, only 2 non-isomorphic τ...

Few families of tournaments satisfying the n-e.c. adjacency property are known. We supply a new random construction for generating infinite families of vertex-transitive n-e.c. tournaments by considering circulant tournaments. Switching is used to generate new ne.c. tournaments of certain orders. With aid of a computer search, we demonstrate that there is a unique minimum order 3-e.c. tournamen...

Problems like the directed feedback vertex set problem have much better algorithms in tournaments when compared to general graphs. This motivates us to study a natural generalization of tournaments, named c-tournaments, and see if the structural properties of these graphs are helpful in obtaining similar algorithms. c-tournaments are simple digraphs which have directed paths of length at most c...

Governments and foundations have successfully harnessed tournaments to spur innovation. Yet this tool is not widely used by firms. We offer a framework for managers seeking to organize tournaments for ideas. We present the theoretical underpinnings of tournaments. We then connect the theory with three recent innovations—the power of the network, the wisdom of crowds, and the power of love—that ...

In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds ho...

The concept of molds, introduced by the authors in a recent preprint, break regular tournaments naturally into big classes: cyclic tournaments, tame tournaments and wild tournaments. We enumerate completely the tame molds, and prove that the dichromatic number of a tame tournament is 3.