Related papers: Hamilton cycles in dense vertex-transitive graphs

In 1963, Anton Kotzig famously conjectured that $K_{n}$, the complete graph of order $n$, where $n$ is even, can be decomposed into $n-1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still…

In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of…

We enumerate certain geometric equivalence classes of subgraphs induced by Hamiltonian paths and cycles in complete graphs. Said classes are orbits under the action of certain direct products of dihedral and cyclic groups on sets of strings…

We first prove a one-to-one correspondence between finding Hamiltonian cycles in a cubic planar graphs and finding trees with specific properties in dual graphs. Using this information, we construct an exact algorithm for finding…

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…

We show that every sufficiently large oriented graph with minimum in- and outdegree at least (3n-4)/8 contains a Hamilton cycle. This is best possible and solves a problem of Thomassen from 1979.

Let $\mathcal{G}(k)$ denote the set of connected $k$-regular graphs $G$, $k\geq2$, where the number of vertices at distance 2 from any vertex in $G$ does not exceed $k$. Asratian (2006) showed (using other terminology) that a graph…

Tutte proved that every 4-connected planar graph contains a Hamilton cycle, but there are 3-connected $n$-vertex planar graphs whose longest cycles have length $\Theta(n^{\log_32})$. On the other hand, Jackson and Wormald in 1992 proved…

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…

A generalized de Bruijn digraph generalizes a de Bruijn digraph to the case where the number of vertices need not be a pure power of an integer. Hamiltonian cycles in these digraphs thus generalize regular de~Bruijn cycles, and we will thus…

We show that $p=\sqrt{\frac{e}{n}}$ is a sharp threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle. This improves the previous results of K\"uhn and Osthus and also Nenadov and \v{S}kori\'c.

We show that the threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle is $p=\frac{1}{\sqrt{n}}$. This improves the previous results of K\"uhn and Osthus and also Nenadov and \v{S}kori\'c. In addition we…

We prove that, for large $n$, every $3$-connected $D$-regular graph on $n$ vertices with $D \geq n/4$ is Hamiltonian. This is best possible and confirms a conjecture posed independently by Bollob\'as and H\"aggkvist in the 1970s. The proof…

We say that a Hamilton cycle $C=(x_1,\ldots,x_n)$ in a graph $G$ is $k$-symmetric, if the mapping $x_i\mapsto x_{i+n/k}$ for all $i=1,\ldots,n$, where indices are considered modulo $n$, is an automorphism of $G$. In other words, if we lay…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

A polynomial time algorithm which detects all paths and cycles of all lengths in form of vertex pairs (start, finish).

Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $\delta n$ edges, where $\delta> 1/2$. Let $\Psi(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic…

A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We…

The P\'osa-Seymour conjecture asserts that every graph on $n$ vertices with minimum degree at least $(1 - 1/(r+1))n$ contains the $r^{th}$ power of a Hamilton cycle. Koml\'os, S\'ark\"ozy and Szemer\'edi famously proved the conjecture for…

A conjecture of Jackson from 1981 states that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large $n$. In fact we prove a more general result that for all…