Related papers: Tournaments with maximal decomposability

Let $R=\bigoplus_{i\geq 0}R_i$ be a Noetherian commutative non-negatively graded ring such that $(R_0,\mathfrak{m}_0)$ is a Henselian local ring. Let $\mathfrak{m}$ be its unique graded maximal ideal $\mathfrak{m}_0+\bigoplus_{i>0}R_i$. Let…

For a regular tournament $T$ of order $n,$ denote by $c_{8}(T)$ the number of cycles of length $8$ in $T.$ Let $DR_{n}$ be a doubly-regular tournament of order $n\equiv 3\mod4$ (so, the out-sets and in-sets of its vertices are also regular…

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 tournament $T$ is a tournament completion of a bipartite tournament $D$ if $D$ is a spanning subdigraph of $T$, i.e., $V(D)=V(T)$ and $A(D)\subseteq A(T)$. If $C$ is a $k$-dicycle (i.e., directed cycle of length $k$) in a tournament…

A digraph $D$ is called \emph{path extendable} if for every nonhamiltonian (directed) path $P$ in $D$, there exists another path $P^\prime$ with the same initial and terminal vertices as $P$, and $V(P^\prime) = V (P)\cup \{w\}$ for a vertex…

The determinant of a tournament $T$ is defined as the determinant of the skew-adjacency matrix of $T$. For a positive odd integer $k$, let $\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $k^2$.…

We prove the following new results. (a) Let $T$ be a regular tournament of order $2n+1\geq 11$ and $S$ a subset of $V(T)$. Suppose that $|S|\leq \frac{1}{2}(n-2)$ and $x$, $y$ are distinct vertices in $V(T)\setminus S$. If the subtournament…

In this thesis we prove a variety of theorems on tournaments. A \emph{prime} tournament is a tournament $G$ such that there is no $X \subseteq V(G)$, $1 < |X| < |V(G)|$, such that for every vertex $v \in V(G) \minus X$, either $v \ra x$ for…

Let $a, \ b \ (b \geq a)$ and $n \ (n \geq 2)$ be nonnegative integers and let $\mathcal{T}(a,b,n)$ be the set of such generalised tournaments, in which every pair of distinct players is connected at most with $b$, and at least with $a$…

We extend the list of tournaments $S$ for which the complete structural description for tournaments excluding $S$ as a subtournament is known. Specifically, let $\Delta(1, 2, 2)$ be a tournament on five vertices obtained from a cyclic…

Let $\vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $\vec{T}_4$'s, and this bound is…

Let $T_{n}$ be an arc-colored tournament of order $n$. The maximum monochromatic indegree $\Delta^{-mon}(T_{n})$ (resp. outdegree $\Delta^{+mon}(T_{n})$) of $T_{n}$ is the maximum number of in-arcs (resp. out-arcs) of a same color incident…

A tournament is a directed graph resulting from an orientation of the complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$ is a matrix with $0$s on its diagonal and all other entries equal to $1$. An outstanding…

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…

We study some problems pertaining to the tournament equilibrium set (TEQ for short). A tournament $H$ is a TEQ-retentive tournament if there is a tournament $T$ which has a minimal TEQ-retentive set $R$ such that $T[R]$ is isomorphic to…

A diamond is a $4$-tournament which consists of a vertex dominating or dominated by a $3$-cycle. Assuming the existence of skew-conference matrices, we give a complete characterization of $n$-tournaments with the maximum number of diamonds…

A $k$-majority tournament $T$ on a finite set of vertices $V$ is defined by a set of $2k-1$ linear orders on $V$, with an edge $u \to v$ in $T$ if $u>v$ in a majority of the linear orders. We think of the linear orders as voter preferences…

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 $c_2=1/2$, $c_3=5/6$…

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…

We consider the manipulability of tournament rules for round-robin tournaments of $n$ competitors. Specifically, $n$ competitors are competing for a prize, and a tournament rule $r$ maps the result of all $\binom{n}{2}$ pairwise matches…