Related papers: Sectionable Tournaments: Their topology and Colori…
Let $G$ be a graph of order $n$. It is well-known that $\alpha(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $\alpha(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs…
Single-elimination (SE) tournaments are a popular format used in competitive environments and decision making. Algorithms for SE tournament manipulation have been an active topic of research in recent years. In this paper, we initiate the…
The orientation completion problem for a class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class by orienting the unoriented edges of the partially oriented graph.…
Continuing the program begun in the first paper of this series, we present a pair of non-reconstructable tournament on $p$ vertices for each $p=2^n \ge 4$.
We determine the inducibility of all tournaments with at most $4$ vertices together with the extremal constructions. The $4$-vertex tournament containing an oriented $C_3$ and one source vertex has a particularly interesting extremal…
A tournament is $k$-spectrally monomorphic if all the $k\times k$ principal submatrices of its adjacency matrix have the same characteristic polynomial. Transitive $n$-tournaments are trivially $k$-spectrally monomorphic. We show that there…
A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and the in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various…
A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph,…
Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations…
We propose a new tournament structure that combines the popular knockout tournaments and the round-robin tournaments. As opposed to the extremes of divisive elimination and no elimination, our tournament aims to eliminate the participants…
A directed graph $R^{\circ}$ on a set $X$ is a set of ordered pairs of distinct points called \emph{arcs}. It is a tournament when every pair of distinct points is connected by an arc in one direction or the other (and not both). We can…
We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(\ell)$ be the limit of the ratio of the maximum number of cycles of length $\ell$ in an $n$-vertex tournament and the…
In the paper, we define a new parameter for tournaments called degreewidth which can be seen as a measure of how far is the tournament from being acyclic. The degreewidth of a tournament $T$ denoted by $\Delta(T)$ is the minimum value $k$…
We extend the open games framework for compositional game theory to encompass also mixed strategies, making essential use of the discrete probability distribution monad. We show that the resulting games form a symmetric monoidal category,…
Transitivity is a central, generative principle in social and other complex networks, capturing the tendency for two nodes with a common neighbor to form a direct connection. We propose a new model for highly dense, complex networks based…
A knockout tournament is one of the most simple and popular forms of competition. Here, we are given a binary tournament tree where all leaves are labeled with seed position names. The players participating in the tournament are assigned to…
Nearly-doubly-regular tournaments have played significant roles in extremal graph theory. In this note, we construct new cyclotomic nearly-doubly-regular tournaments and determine their spectrum by establishing a new connection between…
A hypertree, or $\mathbb{Q}$-acyclic complex, is a higher-dimensional analogue of a tree. We study random $2$-dimensional hypertrees according to the determinantal measure suggested by Lyons. We are especially interested in their…
Ladder tournaments are widely used to rank individuals in real-world organizations and games. Their mathematical properties however are still poorly understood. We formalize the ranking rule generated by a ladder tournament, and we show…
A descent of a labeled digraph is a directed edge (s, t) with s > t. We count strong tournaments, strong digraphs, and acyclic digraphs by descents and edges. To count strong tournaments we use Eulerian generating functions and to count…