Related papers: On tournament inversion

Let $a, b$ and $n$ be nonnegative integers $(b \geq a, \ b > 0, \ n \geq 1)$, $\mathcal{G}_n(a,b)$ be a multigraph on $n$ vertices in which any pair of vertices is connected with at least $a$ and at most $b$ edges and \textbf{v =} $(v_1,…

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 $k$…

A transitive tournament is an acyclic orientation of a complete graph. We study decompositions and packings of the transitive tournament \(TT_n\) into connected two-arc motifs. The three motifs considered are chains, colliders, and forks,…

Let $T$ be a tournament of odd order $n\ge 5,$ $c_{m}(T)$ be the number of its $m$-cycles, and $s_{m}(T)$ be the number of its strongly connected $m$-subtournaments. Due to work of L.W. Beineke and F. Harary, it is well known that…

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…

In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${\rm ex}(T)$ paths, where ${\rm ex}(T):= \frac{1}{2}\sum_{v\in V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for…

A 2.75-approximation algorithm is proposed for the unconstrained traveling tournament problem, which is a variant of the traveling tournament problem. For the unconstrained traveling tournament problem, this is the first proposal of an…

We study biased {\em orientation games}, in which the board is the complete graph $K_n$, and Maker and Breaker take turns in directing previously undirected edges of $K_n$. At the end of the game, the obtained graph is a tournament. Maker…

The theory of tournament limits and tournament kernels (often called graphons) is developed by extending common notions for finite tournaments to this setting; in particular we study transitivity and irreducibility of limits and kernels. We…

We prove that, with high probability, in every $2$-edge-colouring of the random tournament on $n$ vertices there is a monochromatic copy of every oriented tree of order $O (n / \sqrt{\log n})$. This generalises a result of the first, third…

We prove that in a random tournament the events $\{s\rightarrow a\}$ and $\{t\rightarrow b\}$ are positively correlated, for distinct vertices $a,s,b,t \in K_n.$ It is also proven that the correlation between the events $\{s\rightarrow a\}$…

A tournament organizer must select one of $n$ possible teams as the winner of a competition after observing all $\binom{n}{2}$ matches between them. The organizer would like to find a tournament rule that simultaneously satisfies the…

The clique number of a tournament is the maximum clique number of a graph formed by keeping backwards arcs in an ordering of its vertices. We study the time complexity of computing the clique number of a tournament and prove that, for any…

In 1981, Bermond and Thomassen conjectured that for any positive integer $k$, every digraph with minimum out-degree at least $2k-1$ admits $k$ vertex-disjoint directed cycles. In this short paper, we verify the Bermond-Thomassen conjecture…

We study the Maker-Breaker tournament game played on the edge set of a given graph $G$. Two players, Maker and Breaker claim unclaimed edges of $G$ in turns, and Maker wins if by the end of the game she claims all the edges of a pre-defined…

The Traveling Tournament Problem (TTP) is a hard but interesting sports scheduling problem inspired by Major League Baseball, which is to design a double round-robin schedule such that each pair of teams plays one game in each other's home…

Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each…

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ hyperedges and give a construction, using quadratic residues, for an infinite…

Let $T=(V,A)$ be a tournament. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is denoted by $T[X]$. A subset $I$ of $V$ is an interval of $T$ provided that for every $a,b\in I$ and $x\in V\setminus I$, $(a,x)\in A$ if and only…

We study the complexity of counting and finding small tournament patterns inside large tournaments. Given a fixed tournament $T$ of order $k$, we write ${\#}\text{IndSub}_{\text{To}}(\{T\})$ for the problem whose input is a tournament $G$…