Related papers: Sectionable Tournaments: Their topology and Colori…

We show that if $D$ is a tournament of arbitrary size then $D$ has finite strong components after reversing a locally finite sequence of cycles. In turn, we prove that any tournament can be covered by two acyclic sets after reversing a…

A tournament is \emph{acyclically indecomposable} if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically…

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

The \emph{chromatic number} of a directed graph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class of $D$ induces an acyclic subdigraph. Thus, the chromatic number of a tournament $T$ is the…

We prove that every $n$-vertex tournament $G$ has an acyclic subgraph with chromatic number at least $n^{5/9-o(1)}$, while there exists an $n$-vertex tournament $G$ whose every acyclic subgraph has chromatic number at most $n^{3/4+o(1)}$.…

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,…

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…

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…

This paper is a survey of results and problems related to the following question: is it true that if G is a tournament with sufficiently large chromatic number, then G has two vertex-disjoint subtournaments A,B, both with large chromatic…

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…

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its…

A tournament is unimodular if the determinant of its skew-adjacency matrix is $1$. In this paper, we give some properties and constructions of unimodular tournaments. A unimodular tournament $T$ with skew-adjacency matrix $S$ is invertible…

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…

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

We consider a tournament $T=(V, A)$. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is $T[X] = (X, A \cap (X \times X))$. An interval of $T$ is a subset $X$ of $V$ such that for $a, b\in X$ and $ x\in V\setminus X$, $(a,x)\in…

A $k$-coloring of a tournament is a partition of its vertices into $k$ acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a…

Given a tournament T=(V,A), a subset X of $V$ is an interval of T provided that for every a, b in X and x\in V-X, (a,x) in A if and only if (b,x) in A. For example, $\emptyset$, {x}(x in V) and V are intervals of T, called trivial…

It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection…

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…

Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order $\Omega(\frac{n}{\log(n)})$.…