Related papers: Graphs with few Hamiltonian Cycles

Deciding if a graph is a Hamilton graph, also named the Hamilton cycle problem, is important for discrete mathematics and computer science. Due to no characterization to identify Hamilton graphs effectively, there are no tractable…

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$…

Let $\epsilon>0$. We consider the problem of constructing a Hamiltonian graph with $(1+\epsilon)n$ edges in the following controlled random graph process. Starting with the empty graph on $[n]$, at each round a set of $K=K(n)$ edges is…

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 present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a $2$-factor consisting of two chordless cycles, non-hamiltonian cycle permutation graphs and permutation…

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices.…

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 conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at…

In light of Lov\'{a}sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if vertex-transitivity is relaxed, yet a high degree of symmetry--specifically…

It is proven that for any integer $g \ge 0$ and $k \in \{ 0, \ldots, 10 \}$, there exist infinitely many 5-regular graphs of genus $g$ containing a 1-factorisation with exactly $k$ pairs of 1-factors that are perfect, i.e. form a…

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 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…

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph $G=\gc$. In this model $G$ is drawn uniformly from graphs with vertex set $[n]$, $m$ edges and minimum degree at least three. We focus on…

There is a sizable literature on investigating the minimum and maximum numbers of cycles in a class of graphs. However, the answer is known only for special classes. This paper presents a result on the smallest number of cycles in…

Chen, Faudree, Gould, Jacobson, and Lesniak determined the minimum degree threshold for which a balanced $k$-partite graph has a Hamiltonian cycle. We give an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary…

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8…

In 1971, Tutte wrote in an article that "it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian". Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by…

Let $k\geq 2$. We show that, for a sufficiently small $\varepsilon>0$, any sufficiently large $n$-vertex Hamiltonian graph of minimum degree at least $n^{1-\varepsilon}$ contains a $2$-factor consisting of exactly $k$ cycles. This is the…

A non-planar graph is almost-planar if either deleting or contracting any edge makes it planar. A graph with $n$ vertices is pancyclic if it contains a cycle of every length from $3$ to $n$, and it is Hamiltonian if it contains a cycle of…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…