Related papers: Ore-degree threshold for the square of a Hamiltoni…

Let $G$ be a Hamiltonian graph with $n$ vertices. A nonempty vertex set $X\subseteq V(G)$ is called a Hamiltonian cycle enforcing set (in short, an $H$-force set) of $G$ if every $X$-cycle of $G$ (i.e., a cycle of $G$ containing all…

We conjecture that every oriented graph $G$ on $n$ vertices with $\delta ^+ (G) , \delta ^- (G) \geq 5n/12$ contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing…

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. In 1972, Erd\H{o}s conjectured that every Hamiltonian graph with…

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[…

We show that every sufficiently large oriented graph $G$ with minimum indegree and outdegree both at least $(3|V(G)|-1)/8$ contains every orientation of a Hamilton cycle. This result improves the approximate bound established by Kelly and…

In 1960, Ghouila-Houri proved that every strongly connected directed graph $G$ on $n$ vertices with minimum degree at least $n$ contains a directed Hamilton cycle. We asymptotically generalize this result by proving the following: every…

The study of graph discrepancy problems, initiated by Erd\H{o}s in the 1960s, has received renewed attention in recent years. In general, given a $2$-edge-coloured graph $G$, one is interested in embedding a copy of a graph $H$ in $G$ with…

We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $\mu n$ for arbitrarily small $\mu>0$. About 20 years ago Koml\'os, Sark\"ozy, and Szemer\'edi resolved…

In 2006, K\"{u}hn and Osthus showed that if a 3-graph H on n vertices has minimum co-degree at least (1/4 +o(1))n and n is even then H has a loose Hamilton cycle. In this paper, we prove that the minimum co-degree of n/4 suffices. The…

Let $k \geq 3$ be an integer, $h_{k}(G)$ be the number of vertices of degree at least $2k$ in a graph $G$, and $\ell_{k}(G)$ be the number of vertices of degree at most $2k-2$ in $G$. Dirac and Erd\H{o}s proved in 1963 that if $h_{k}(G) -…

Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G)\colon xy\in E(G)\}$. The minimum common degree of $G$, denoted by $\delta_{2}(G)$, is defined as the minimum of $|N(x)\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982,…

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…

Consider the random subgraph process on a base graph $G$ with $n$ vertices: we generate a sequence $\{G_t\}_{t=0}^{|E(G)|}$ by taking a uniformly random ordering of the edges of $G$ and then adding these edges one by one to the empty graph…

We study Hamiltonicity in the union of an $n$-vertex graph $H$ with high minimum degree and a binomial random graph on the same vertex set. In particular, we consider the case when $H$ has minimum degree close to $n/2$. We determine the…

For any $k\ge 3$ and $\ell \in [k-1]$ such that $(k,\ell) \ne (3,1)$, we show that any sufficiently large $k$-graph $G$ must contain a Hamilton $\ell$-cycle provided that it has no isolated vertices and every set of $k-1$ vertices contained…

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains…

Ore in 1961 determined the maximum number of edges in graphs not containing a Hamiltonian cycle, and Tur\'{a}n in 1941 found the maximum number of edges in graphs not containing a $K_{r+1}$. Motivated by the work of Adamus in 2009 and…

Dirac introduced the notion of a k-critical graph, a graph that is not (k-1)-colorable but whose every proper subgraph is (k-1)-colorable. Brook's Theorem states that every graph with maximum degree k is k-colorable unless it contains a…

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest…