Related papers: On $d$-panconnected tournaments with large semideg…
We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples. We also prove some results on the existence of large packings of $k$-vertex transitive tournaments in an $n$-vertex tournament.…
In this paper, we study $(1,2)$-step competition graphs of bipartite tournaments. A bipartite tournament means an orientation of a complete bipartite graph. We show that the $(1,2)$-step competition graph of a bipartite tournament has at…
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…
We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by K\"uhn…
We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an {\bf out-colouring}. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a…
We study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a…
The celebrated Erd\"{o}s-Hajnal conjecture states that for every undirected graph $H$ there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or…
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…
Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every…
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…
A pure pair in a tournament $G$ is an ordered pair $(A,B)$ of disjoint subsets of $V(G)$ such that every vertex in $B$ is adjacent from every vertex in $A$. Which tournaments $H$ have the property that if $G$ is a tournament not containing…
The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph $H$ there exists a constant $c(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least…
Real world tournaments are almost always intransitive. Recent works have noted that parametric models which assume $d$ dimensional node representations can effectively model intransitive tournaments. However, nothing is known about the…
We prove that in a random tournament the events $\{s\rightarrow a\}$ and $\{t\rightarrow b\}$ are positively correlated, for distinct vertices $a,s,b,t \in K_n.$ It is also proven that the correlation between the events $\{s\rightarrow a\}$…
The score set of a tournament is defined as the set of its distinct out-degrees. In 1978, Reid proposed the conjecture that for any set of nonnegative integers $D$, there exists a tournament $T$ with a degree set $D$. In 1989, Yao presented…
A 3-tournament is a complete 3-uniform hypergraph where each edge has a special vertex designated as its tail. A vertex set $X$ dominates $T$ if every vertex not in $X$ is contained in an edge whose tail is in $X$. The domination number of…
A shortcut of a directed path $v_1 v_2 \cdots v_n$ is an edge $v_iv_j$ with $j > i+1$. If $j = i+2$ the shortcut is called a hop. If all hops are present, the path is called hop complete, so the path and its hops form a square of a path. We…
Using a switching operation on tournaments we obtain some new lower bounds on the Tur\'{a}n number of the $r$-graph on $r+1$ vertices with $3$ edges. For $r=4$, extremal examples were constructed using Paley tournaments in previous work. We…
Erd\"{o}s-Hajnal conjecture states that for every undirected graph $H$ there exists $ \epsilon(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of…
Let $D$ be an directed graph on $p\geq 10$ vertices with minimum degree at least $p-1$ and minimum semi-degree at least $ p/2 -1$. We present a detailed proof of the following result [13]: The digraph $D$ is pancyclic, unless some extremal…