Related papers: Hamilton cycles in a semi-random graph model

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 Hamilton cycle in a directed graph $G$ is a cycle that passes through every vertex of $G$. A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. In the late $60$s Kelly conjectured that every…

Given graphs $G_1,\ldots,G_s$ all on a common vertex set and a graph $H$ with $e(H) = s$, a copy of $H$ is \emph{transversal} or \emph{rainbow} if it contains one edge from each $G_i$. We establish a stability result for transversal…

Counting the number of Hamiltonian cycles that are contained in a geometric graph is {\bf \#P}-complete even if the graph is known to be planar \cite{lot:refer}. A relaxation for problems in plane geometric graphs is to allow the geometric…

We present exponential and super factorial lower bounds on the number of Hamiltonian cycles passing through any edge of the basis graphs of a graphic, generalized Catalan and uniform matroids. All lower bounds were obtained by a common…

We study conditions under which a given hypergraph is randomly robust Hamiltonian, which means that a random sparsification of the host graph contains a Hamilton cycle with high probability. Our main contribution provides nearly optimal…

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…

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n. For any constant a > 0, we give an optimal answer in the following sense: let…

We consider the geometric random (GR) graph on the $d-$dimensional torus with the $L_\sigma$ distance measure ($1 \leq \sigma \leq \infty$). Our main result is an exact characterization of the probability that a particular labeled cycle…

We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair…

Given a graph $G$ and an $r$-edge-colouring $\chi$ on $E(G)$, a Hamilton cycle $H\subset G$ is said to have $t$ colour-bias if $H$ contains $n/r+t$ edges of the same colour in $\chi$. Freschi, Hyde, Lada and Treglown showed every…

Let $\mu > 2$ and $\epsilon > 0$. We show that, if $G$ is a sufficiently large simple graph of average degree at least $\mu$, and $H$ is a random spanning subgraph of $G$ formed by including each edge independently with probability $p \ge…

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…

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to…

Finding Hamitonian Cycles in square grid graphs is a well studied and important questions. More recent work has extended these results to triangular and hexagonal grids, as well as further restricted versions. In this paper, we examine a…

We first consider the following problem. We are given a fixed perfect matching $M$ of $[n]$ and we add random edges one at a time until there is a Hamilton cycle containing $M$. We show that w.h.p. the hitting time for this event is the…

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…

A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{loose Hamilton cycle} if $C$ covers all the vertices of $H$ and there exists a cyclic ordering of these vertices such that the edges in $C$ are segments of that order and such…

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…

In 1999, Katona and Kierstead conjectured that if a $k$-uniform hypergraph $\cal H$ on $n$ vertices has minimum co-degree $\lfloor \frac{n-k+3}{2}\rfloor$, i.e., each set of $k-1$ vertices is contained in at least $\lfloor…