Related papers: Resilience for tight Hamiltonicity

Let $n$ and $r$ be integers with $n-2\ge r\ge 3$. We prove that any $r$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with spectral radius $\lambda(\mathcal{H}) > \binom{n-2}{r-1}$ must contain a Hamiltonian Berge cycle unless…

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…

A covering of a digraph $D$ by Hamilton cycles is a collection of directed Hamilton cycles (not necessarily edge-disjoint) that together cover all the edges of $D$. We prove that for $1/2 \geq p\geq \frac{\log^{20} n}{n}$, the random…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion…

This MSci thesis surveys results in extremal graph theory, in particular relating to Hamilton cycles. Szem\'eredi's Regularity Lemma plays a central role. We also investigate the robust outexpansion property for digraphs. Kelly showed that…

Let $\{D_M\}_{M\geq 0}$ be the $n$-vertex random directed graph process, where $D_0$ is the empty directed graph on $n$ vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly…

A graph is called Dirac if its minimum degree is at least half of the number of vertices in it. Joos and Kim showed that every collection $\mathbb{G}=\{G_1,\ldots,G_n\}$ of Dirac graphs on the same vertex set $V$ of size $n$ contains a…

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 the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.

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…

Chen et al. proved that every 18-tough chordal graph has a Hamilton cycle [Networks 31 (1998), 29-38]. Improving upon their bound, we show that every 10-tough chordal graph is Hamiltonian (in fact, Hamilton-connected). We use Aharoni and…

In this paper, we consider a random geometric graph (RGG)~\(G\) on~\(n\) nodes with adjacency distance~\(r_n\) just below the Hamiltonicity threshold and construct Hamiltonian cycles using additional edges called bridges. The bridges by…

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…

The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ denotes the subspace of $C(G)$, spanned by the…

Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters, yet the existence of many of them is still under the question. In this paper, we continue the study of the famuly of strongly regular…

We prove that for every $\varepsilon > 0$ there is $c_0$ such that if $G\sim G(n,c/n)$, $c\ge c_0$, then with high probability $G$ can be covered by at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ vertex disjoint paths, which is…

We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy…

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum…

If $G$ is a more than one tough graph on $n$ vertices with $\delta\ge \frac{n}{2}-a$ for a given $a>0$ and $n$ is large enough then $G$ is hamiltonian.