Related papers: Tur\'{a}n numbers and switching

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…

It is proved that for integers $b, r$ such that $3 \leq b < r \leq \binom{b+1}{2} - 1$, there exists a red/blue edge-colored graph such that the red degree of every vertex is $r$, the blue degree of every vertex is $b$, yet in the closed…

The classic Rock-Paper-Scissors game of size 3 and its extension, Rock-Paper-Scissors-Lizard-Spock, are modeled by directed graphs called tournaments. They can be further extended to any odd size. The extended games are regular tournaments…

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 $n^{3/4+o(1)}$.…

We study variants of Sidorenko's conjecture in tournaments, where new phenomena arise that do not have clear analogues in the setting of undirected graphs. We first consider oriented graphs that are systematically under-represented in…

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…

The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \: n^r \:…

Suppose one needs to change the direction of at least $\epsilon n^2$ edges of an $n$-vertex tournament $T$, in order to make it $H$-free. A standard application of the regularity method shows that in this case $T$ contains at least…

We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for…

A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it…

Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\'{a}n graph on $n$ vertices with a maximum clique of order…

Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at…

The Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free graph on $n$ vertices. Let $T$ be any tree. The odd-ballooning of $T$, denoted by $T_o$, is a graph obtained by replacing each edge of $T$ with an odd cycle…

The numbers game is a one-player game played on a finite simple graph with certain ``amplitudes'' assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at…

Based on known methods for computing the number of distinct score sequences for $n$-vertex tournaments, we develop algorithms for computing the number of distinct score sequences for self-complementary tournaments, strong tournaments, and…

We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_t$ of colours (independent of the past) from $n$ colours of balls, and then chooses a colour $i$ from this subset with…

Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all $n \ge 6$, the maximum…

The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. Bessy, Sereni and Lichiardopol proved that a regular…

A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called *kings* of the…

Given non-negative integers $n_{i}$ and $\alpha_{i}$ with $0 \leq \alpha_{i} \leq n_i$ $(i=1,2,...,k)$, an $[\alpha_{1},\alpha_{2},...,\alpha_{k}]$-$k$-partite hypertournament on $\sum_{1}^{k}n_{i}$ vertices is a $(k+1)$-tuple…