Related papers: Hamilton transversals in tournaments

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on…

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…

Given a collection $\mathcal{D} =\{D_1,D_2,\ldots,D_m\}$ of digraphs on the common vertex set $V$, an $m$-edge digraph $H$ with vertices in $V$ is \textit{transversal} in $\mathcal{D}$ if there exists a bijection $\varphi :E(H)\rightarrow…

We use a randomised embedding method to prove that for all \alpha>0 any sufficiently large oriented graph G with minimum in-degree and out-degree \delta^+(G),\delta^-(G)\geq (3/8+\alpha)|G| contains every possible orientation of a Hamilton…

A Hamilton cycle in a directed graph $G$ is a cycle that passes through every vertex of $G$. A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. In the late $60$s Kelly conjectured that every…

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…

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…

A tuple $(G_1,\dots,G_n)$ of graphs on the same vertex set of size $n$ is said to be Hamilton-universal if for every map $\chi: [n]\to[n]$ there exists a Hamilton cycle whose $i$-th edge comes from $G_{\chi(i)}$. Bowtell, Morris, Pehova and…

The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive…

A multipartite tournament is an orientation of a complete $c$-partite graph. In [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007) 1148--1150], Volkmann proved that a…

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…

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…

It was shown by Kutnar and \v Sparl in 2009 that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except…

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…

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly…

The dominating graph of a graph $H$ has as its vertices all dominating sets of $H$, with an edge between two dominating sets if one can be obtained from the other by the addition or deletion of a single vertex of $H$. In this paper we prove…

It is shown that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of…

Let $\mathbf{G}=\{G_1,\dots,G_{s}\}$ be a collection of $s$ bipartite graphs with the same bipartition $V=(X,Y)$. For a path $P$ with $V(P)=V$ and $|E(P)|=s$, if there exists an injection $\phi$: $E(P)\rightarrow [s]$ such that $e\in…

A step forward is made in a long standing Lov\'{a}sz's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph,…

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…