Related papers: No additional tournaments are quasirandom-forcing
A sequence $\pi_1,\pi_2,\dots$ of permutations is said to be "quasirandom" if the induced density of every permutation $\sigma$ in $\pi_n$ converges to $1/|\sigma|!$ as $n\to\infty$. We prove that $\pi_1,\pi_2,\dots$ is quasirandom if and…
K\"uhn, Osthus, and Townsend asked whether there exists a constant $C$ such that every strongly $Ct$-connected tournament contains all possible $1$-factors with at most $t$ components. We answer this question in the affirmative. This is…
An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical'…
We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families $\\mathcal F$ of graphs with the property that if a large graph $G$ has approximately homomorphism density $p^{e(F)}$ for some fixed…
A subset $S$ of initially infected vertices of a graph $G$ is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects…
A tournament is said to have the $S_k$-property if, for any set of $k$ players, there is another player who beats them all. Minimum tournaments having this property have been explored very well in the 1960's and the early 1970's. In this…
In his paper "Kings in Bipartite Hypertournaments" (Graphs $\&$ Combinatorics 35, 2019), Petrovic stated two conjectures on 4-kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.
A Walecki tournament is any tournament that can be formed by choosing an orientation for each of the Hamilton cycles in the Walecki decomposition of a complete graph on an odd number of vertices. In this paper, we show that if some arc in a…
A digraph is {\bf eulerian} if it is connected and every vertex has its in-degree equal to its out-degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. A digraph is {\bf semicomplete} if it has no…
A tournament on $n$ agents is a complete oriented graph with the agents as vertices and edges that describe the win-loss outcomes of the $\binom{n}{2}$ matches played between each pair of agents. The winner of a tournament is determined by…
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 further study sets of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on sets with an additional symmetry we call "balance," we prove that sets of $n$ such $m$-sided dice exist for all $n,m \geq 3$.…
A result of Simonovits and S\'os states that for any fixed graph $H$ and any $\epsilon > 0$ there exists $\delta > 0$ such that if $G$ is an $n$-vertex graph with the property that every $S \subseteq V(G)$ contains $p^{e(H)} |S|^{v(H)} \pm…
We show that if $T$ is a strongly $10^9k^6\log(2k)$-connected tournament, there exists a partition $A, B$ of $V(T)$ such that each of $T[A]$, $T[B]$ and $T[A,B]$ is strongly $k$-connected. This provides tournament analogues of two partition…
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to…
We study the effects of randomness on competitions based on an elementary random process in which there is a finite probability that a weaker team upsets a stronger team. We apply this model to sports leagues and sports tournaments, and…
Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of…
The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…
A generalized $N$-sided die is a random variable $D$ on a sample space of $N$ equally likely outcomes taking values in the set of positive integers. We say of independent $N$ sided dice $D_i, D_j$ that $D_i$ beats $D_j$, written $D_i \to…
The sequence counting the number of unique tournaments with n people is the same as the sequence counting non-tracking binary strings corresponding to n-2 radar observations with the tracking rule "3 out of 5 with loss 2." This fact allows…