Related papers: On $d$-panconnected tournaments with large semideg…

A $k$-hypertournament $H$ on $n$ vertices is a pair $(V(H),A(H))$, where $V(H)$ is a set of vertices and $A(H)$ is a set of $k$-tuples of vertices, called arcs, such that for any $k$-subset $S$ of $V(H)$, $A(H)$ contains exactly one of the…

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…

Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2 \rightarrow v_1$, and $v_i \rightarrow v_j$ if $j-i \equiv 1$, $2 \pmod{5}$ and ${i,j} \neq {1,2}$. In this paper we describe the tournaments which do not have $U_5$…

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 tournament on 8 or more vertices may be intrinsically linked as a directed graph. We begin the classification of intrinsically linked tournaments by examining their score sequences. While many distinct tournaments may have the same score…

Given a tournament $T$, a module of $T$ is a subset $X$ of $V(T)$ such that for $x, y\in X$ and $v\in V(T)\setminus X$, $(x,v)\in A(T)$ if and only if $(y,v)\in A(T)$. The trivial modules of $T$ are $\emptyset$, $\{u\}$ $(u\in V(T))$ and…

Let $T$ be a strongly connected tournament of order $n\ge 4$ whose diameter does not exceed $d\ge 3.$ Denote by $c_{\ell}(T)$ the number of circuits of length $\ell$ in $T.$ In our recent paper, we construct a strongly connected tournament…

Continuing the program begun in the first paper of this series, we present a pair of non-reconstructable tournament on $p$ vertices for each $p=2^n \ge 4$.

A multipartite tournament is an orientation of a complete $c$-partite graph. In [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007) 1148--1150], Volkmann proved that a…

Let $k$ be a positive integer. Let $G$ be a balanced bipartite graph of order $2n$ with bipartition $(X, Y)$, and $S$ a subset of $X$. Suppose that every pair of nonadjacent vertices $(x,y)$ with $x\in S, y\in Y$ satisfies $d(x)+d(y)\geq…

An oriented graph $\vec{H}$ is said to be tournament anti-Sidorenko if the homomorphism density of $\vec{H}$ in any tournament $\vec{T}$ is bounded above by the homomorphism density of $\vec{H}$ in a large uniformly random tournament. We…

Let $D$ be a digraph. We define the minimum semi-degree of $D$ as $\delta^{0}(D) := \min \{\delta^{+}(D), \delta^{-}(D)\}$. Let $k$ be a positive integer, and let $S = \{s\}$ and $T = \{t_{1}, \dots ,t_{k}\}$ be any two disjoint subsets of…

We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: (i) If in $T$ one cannot find…

A tournament $H$ is said to force quasirandomness if it has the property that a sequence $(T_n)_{n\in \mathbb{N}}$ of tournaments of increasing orders is quasirandom if and only if the homomorphism density of $H$ in $T_n$ tends to…

A vertex $x$ in a tournament $T$ is called a king if for every vertex $y$ of $T$ there is a directed path from $x$ to $y$ of length at most 2. It is not hard to show that every vertex of maximum out-degree in a tournament is a king.…

We consider the transformation reversing all arcs of a subset $X$ of the vertex set of a tournament $T$. The \emph{index} of $T$, denoted by $i(T)$, is the smallest number of subsets that must be reversed to make $T$ acyclic. It turns out…

An {\it inversion} of a tournament $T$ is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let ${\rm inv}_k(T)$ be the minimum length of a sequence of inversions using sets of size at most $k$…

In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V_1,...,V_t such that for all i the subtournament T[V_i]…

Let $TT_k$ denote the transitive tournament on $k$ vertices. Let $TT(h,k)$ denote the graph obtained from $TT_k$ by replacing each vertex with an independent set of size $h \geq 1$. The following result is proved: Let $c_2=1/2$, $c_3=5/6$…

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented…