Related papers: Compatible Hamilton cycles in Dirac graphs

A famous theorem of Dirac states that any graph on $n$ vertices with minimum degree at least $n/2$ has a Hamilton cycle. Such graphs are called Dirac graphs. Strengthening this result, we show the existence of rainbow Hamilton cycles in…

Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering…

A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $\lceil n/2 \rceil$ is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of…

Dirac proved that any graph with minimum vertex degree $\delta$ contains either a cycle of length at least $2\delta$ or a Hamilton cycle. Motivated by this result, we characterize those graphs having no cycle longer than $2\delta$.

A well known generalisation of Dirac's theorem states that if a graph $G$ on $n\ge 4k$ vertices has minimum degree at least $n/2$ then $G$ contains a $2$-factor consisting of exactly $k$ cycles. This is easily seen to be tight in terms of…

In 1970 Lov\'asz conjectured that every connected vertex-transitive graph admits a Hamilton cycle, apart from five exceptional graphs. This conjecture has recently been settled for graphs defined by intersecting set systems, which feature…

Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\delta(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex…

In this paper we consider the existence of Hamilton cycles in the random graph $G=G_{n,m}^{\delta\geq 3}$. This a random graph chosen uniformly from the set of graphs with vertex set $[n]$, $m$ edges and minimum degree at least 3. Our…

A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$…

We show that for each \alpha>0 every sufficiently large oriented graph G with \delta^+(G),\delta^-(G)\ge 3|G|/8+ \alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. In fact, we prove the…

For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We…

A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order.…

A Hamilton cycle is a cycle containing every vertex of a graph. A graph is called Hamiltonian if it contains a Hamilton cycle. The Hamilton cycle problem is to find the sufficient and necessary condition that a graph is Hamiltonian. In this…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there…

Let $G$ be a graph on $n\geq 3$ vertices, claw the bipartite graph $K_{1,3}$, and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$ to its one vertex. $G$ is called 1-heavy if at least one end vertex of each induced…

The famous Dirac's Theorem gives an exact bound on the minimum degree of an $n$-vertex graph guaranteeing the existence of a hamiltonian cycle. We prove exact bounds of similar type for hamiltonian Berge cycles in $r$-uniform, $n$-vertex…

Let $D$ be a strongly connected directed graph of order $n\geq 4$ which satisfies the following condition (*): for every pair of non-adjacent vertices $x, y$ with a common in-neighbour $d(x)+d(y)\geq 2n-1$ and $min \{ d(x), d(y)\}\geq n-1$.…

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved…

For a collection $\mathbf{G}=\{G_1,\dots, G_s\}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $\mathbf{G}$-transversal if there exists a bijection $\phi:E(H)\rightarrow [s]$ such that…