Related papers: A proof of Sumner's universal tournament conjectur…
Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains a copy of any directed tree on $n$ vertices. We prove an asymptotic version of this conjecture, namely that any tournament on $(2+o(1))n$…
Sumner's universal tournament conjecture states that every $(2n-2)$-vertex tournament should contain a copy of every $n$-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a…
An oriented tree $T$ on $n$ vertices is unavoidable if every tournament on $n$ vertices contains a copy of $T$. In this paper we give a sufficient condition for $T$ to be unavoidable, and use this to prove that almost all labelled oriented…
We prove that, with high probability, in every $2$-edge-colouring of the random tournament on $n$ vertices there is a monochromatic copy of every oriented tree of order $O (n / \sqrt{\log n})$. This generalises a result of the first, third…
Seymour's distance two conjecture states that in any digraph there exists a vertex (a "Seymour vertex") that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements…
A celebrated unresolved conjecture of Erd\"{o}s and Hajnal 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…
A celebrated unresolved conjecture of Erd\H{o}s and Hajnal 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…
In this note we propose a generalisation of Seymour's Second Neighbourhood Conjecture to two directed graphs on a vertex set. We prove that this generalisation holds in the case of tournaments, and we show that a natural strengthening of…
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…
Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. We prove its weighted version for tournaments missing a…
Sullivan stated the conjectures: (1) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)\geq d^{-}(x)$; (2) every oriented graph $D$ has a vertex $x$ such that $d^{++}(x)+d^{+}(x)\geq 2d^{-}(x)$. In this paper, we prove that…
We prove that there exists $C>0$ such that any $(n+Ck)$-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves, improving the previously best known bound of $n+O(k^2)$ vertices to give a result tight up to the…
A celebrated unresolved conjecture of Erd\"{o}s and Hajnal 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…
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…
A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erd\H{o}s-Hajnal Conjecture, states that for every tournament $S$ there exists $\epsilon(S)>0$ such that if $T$ is an $n$-vertex tournament that does not…
Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This…
Rosenfeld Conjectured in 1972 that there exists an integer K $\geq$ 8 such that any tournament of order n $\geq$ K contains any Hamiltonian oriented path. In 2000, Havet and Thomass\'e proved this conjecture for any tournament with exactly…
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…
In this work we present a version of the so called Chen and Chv\'atal's conjecture for directed graphs. A line of a directed graph D is defined by an ordered pair (u, v), with u and v two distinct vertices of D, as the set of all vertices w…
In this note we show that every tournament on $n$ vertices contains the $k$-th power of a directed path of length $n/2^{6k+7}$, which improves upon the recent bound of Scott and Kor\'{a}ndi of $n/2^{2^{3k}}$. By doing so, we get an inverse…