Related papers: Loose Hamilton Cycles in Random 3-Uniform Hypergra…

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 \[…

A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Broersma, Patel, and Pyatkin showed that every 25-tough $2K_2$-free graph with at least three vertices is hamiltonian. In this paper, we…

We show that every 3-uniform hypergraph with $n$ vertices and minimum vertex degree at least $(5/9+o(1))\binom{n}2$ contains a tight Hamiltonian cycle. Known lower bound constructions show that this degree condition is asymptotically…

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…

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…

We revisit the method of small subgraph conditioning, used to establish that random regular graphs are Hamiltonian a.a.s. We refine this method using new technical machinery for random $d$-regular graphs on $n$ vertices that hold not just…

Fix a positive integer $n$, a real number $p\in (0,1]$, and a (perhaps random) hypergraph $\mathcal{H}$ on $[n]$. We introduce and investigate the following random multigraph model, which we denote $\mathbb{G}(n,p\, ; \,\mathcal{H})$: begin…

Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its…

Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. R\"{o}dl, A. %Ruci\'{n}ski, M. Simonovits, J. Skokan, The Ramsey number for hypergraph cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color Ramsey number of…

We consider how many random edges need to be added to a graph of order $n$ with minimum degree $\alpha n$ in order that it contains the square of a Hamilton cycle w.h.p..

It is proved that if $G$ is a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$ then either $G$ has a cycle of length at least $\min\{n,2\delta+4\}$ or $G$ is the Petersen graph.

Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that…

We prove that if an $n$-vertex graph with minimum degree at least $3$ contains a Hamiltonian cycle, then it contains another cycle of length $n-o(n)$; this implies, in particular, that a well-known conjecture of Sheehan from 1975 holds…

We study the existence of directed Hamilton cycles in random digraphs with $m$ edges where we condition on minimum in- and out-degree $\d \ge k+1$, where $k \ge 1$. Denote such a random graph by $D_{n,m}^{(\delta\geq k+1)}$. Let $m=cn$ and…

Let integer $n \ge 3$ and integer $r = r(n) \ge 3$. Define the binomial random $r$-uniform hypergraph $H_r(n, p)$ to be the $r$-uniform graph on the vertex set $[n]$ such that each $r$-set is an edge independently with probability $p$. A…

Investigating a problem of B. Mohar, we show that every one-ended Hamiltonian cubic graph with end degree 3 contains a second Hamilton cycle. We also construct two examples showing that this result does not extend to give a third Hamilton…

Visibility graph of a simple polygon is a graph with the same vertex set in which there is an edge between a pair of vertices if and only if the segment through them lies completely inside the polygon. Each pair of adjacent vertices on the…

Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain…

A Hamiltonian path (cycle) in a graph is a path (cycle, respectively) which passes through all of its vertices. The problems of deciding the existence of a Hamiltonian cycle (path) in an input graph are well known to be NP-complete, and…

Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markstr\"{o}m asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such…