Related papers: Finding tight Hamilton cycles in random hypergraph…

We are given a graph $G$ with $n$ vertices, where a random subset of $k$ vertices has been made into a clique, and the remaining edges are chosen independently with probability $\tfrac12$. This random graph model is denoted…

We establish an inclusion relation between two uniform models of random $k$-graphs (for constant $k \ge 2$) on $n$ labeled vertices: $\mathbb G^{(k)}(n,m)$, the random $k$-graph with $m$ edges, and $\mathbb R^{(k)}(n,d)$, the random…

We prove that for integers $2 \leq \ell < k$ and a small constant $c$, if a $k$-uniform hypergraph with linear minimum codegree is randomly `perturbed' by changing non-edges to edges independently at random with probability $p \geq…

A {\it weak (Berge) cycle} is an alternating sequence of vertices and (hyper)edges $C=(v_0, e_1, v_1, ..., v_{\ell-1}, e_\ell, v_{\ell}=v_0)$ such that the vertices $v_0, ..., v_{\ell-1}$ are distinct with $v_k, v_{k+1} \in e_{k}$ for each…

Let $H$ be an edge colored hypergraph. We say that $H$ contains a \emph{rainbow} copy of a hypergraph $S$ if it contains an isomorphic copy of $S$ with all edges of distinct colors. We consider the following setting. A randomly edge colored…

Given a graph on $n$ vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length $n$ visiting each vertex once and with pairwise different colours on the edges. Similarly (for even $n$) a rainbow…

Finding near-rainbow Hamilton cycles in properly edge-coloured graphs was first studied by Andersen, who proved in 1989 that every proper edge colouring of the complete graph on $n$ vertices contains a Hamilton cycle with at least…

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…

For integers $k \geq 3$ and $r\geq 2$, we show that for every $\alpha> 0$, there exists $\varepsilon > 0$ such that the union of $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\alpha n$ and a binomial random…

We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and random directed graphs, D. If n is finite, we assume that G or D contains a hamilton circuit. If G is an arbitrary graph containing a hamilton…

We present a deterministic algorithm that given any directed graph on n vertices computes the parity of its number of Hamiltonian cycles in O(1.619^n) time and polynomial space. For bipartite graphs, we give a 1.5^n poly(n) expected time…

In order to find Hamiltonian cycle, algorithm should find edges that creates a Hamiltonian cycle. Higher number of edges creates more possibilities to check to solve the problem. Algorithm rests on analysis of original graph and opposite…

A circulant graph H is defined on the set of vertices V=\left\{ 1,\ldots,n\right\} and edges E=\left\{ \left(i,j\right):\left|i-j\right|\equiv s\left(\textrm{mod}n\right),s\in S\right\} , where S\subseteq\left\{…

We propose an improved algorithm for counting the number of Hamiltonian cycles in a directed graph. The basic idea of the method is sequential acceptance/rejection, which is successfully used in approximating the number of perfect matchings…

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…

A large number of NP-hard graph problems can be solved in $f(w)n^{O(1)}$ time and space when the input graph is provided together with a tree decomposition of width $w$, in many cases with a modest exponential dependence $f(w)$ on $w$.…

We present a Monte Carlo algorithm for Hamiltonicity detection in an $n$-vertex undirected graph running in $O^*(1.657^{n})$ time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the…

For positive integers $r > \ell \geq 1$, an $\ell$-cycle in an $r$-uniform hypergraph is a cycle where each edge consists of $r$ vertices and each pair of consecutive edges intersect in $\ell$ vertices. For $\ell \geq 2$, we determine the…

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 present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large ``bipartite hole`` (two disjoint sets of vertices with no edges between them). This result extends Dirac's…