Related papers: Paths in tournaments a simple proof of Rosenfeld's…
Rosenfeld in 1974 conjectured that there is an integer N > 8 such that every tournament of order n > N contains every non-directed cycle of order n. We prove that, with exactly 35 exceptions, every tournament of order n > 2 contains each…
We prove that a tournament and its complement contain the same number of oriented Hamiltonian paths (resp. cycles) of any given type, as a generalization of Rosenfeld's result proved for antidirected paths.
Havet and Thomass\'{e} proved that every tournament of order $n\geq 8$ contains every oriented Hamiltonian path, which was conjectured by Rosenfeld. Recently, it was shown that in any tournament $T$ of order $n\geq 8$, there exists an arc…
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…
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…
A conjecture of Thomassen from 1982 states that for every k there is an f(k) so that every strongly f(k)-connected tournament contains k edge-disjoint Hamilton cycles. A classical theorem of Camion, that every strongly connected tournament…
We prove that, with high probability, any $2$-edge-colouring of a random tournament on $n$ vertices contains a monochromatic path of length $\Omega(n / \sqrt{\log n})$. This resolves a conjecture of Ben-Eliezer, Krivelevich and Sudakov and…
We show that for every positive integer $k$, any tournament can be partitioned into at most $2^{ck}$ $k$-th powers of paths. This result is tight up to the exponential constant. Moreover, we prove that for every $\varepsilon>0$ and every…
In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${\rm ex}(T)$ paths, where ${\rm ex}(T):= \frac{1}{2}\sum_{v\in V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for…
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…
Brandt et al. (2013) have recently disproved a conjecture by Schwartz (1990) by non-constructively showing the existence of a counterexample with about 10^136 alternatives. We provide a concrete counterexample for Schwartz's conjecture with…
In 1934 L. R\'edei published his famous theorem that the number of Hamiltonian paths in a tournament is odd. In fact it is a corollary of a stronger theorem in his paper. Stronger theorems were also obtained in the early 1970s by G.A. Dirac…
We construct a combinatorial function F which computes the number of oriented Hamiltonian paths of any given type, in a transitive tournament. We also study many properties of F that arise, and reach some observations.
In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of…
A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be…
An open conjecture of Erd\H{o}s states that for every positive integer $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so…
Let $H$ be an oriented graph without directed cycle. The oriented Ramsey number of $H$, denoted by $\overrightarrow{r}(H)$, is the smallest integer $N$ such that every tournament on $N$ vertices contains a copy of $H$. Rosenfeld (JCT-B,…
Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains any directed tree on $n$ vertices. In this paper we prove that this conjecture holds for all sufficiently large $n$. The proof makes extensive…
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…
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…