Related papers: On Hamilton cycles in graphs defined by intersecti…
A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such…
A step forward is made in a long standing Lov\'{a}sz's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph,…
We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…
In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two…
Lov\'{a}sz conjectured that every connected vertex-transitive graph contains a hamilton path in 1970. First we reveal the structure of connected vertex-transitive graphs with an odd number of vertices. Then we prove that every connected…
A Hamilton cycle is a cycle containing every vertex of a graph. A graph is called Hamiltonian if it contains a Hamilton cycle. The Hamilton cycle problem is to find the sufficient and necessary condition that a graph is Hamiltonian. In this…
It was shown by Kutnar and \v Sparl in 2009 that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except…
After long term efforts, it was recently proved in \cite{DKM2} that except for the Peterson graph, every connected vertex-transitive graph of order $rs$ has a Hamilton cycle, where $r$ and $s$ are primes. A natural topic is to solve the…
A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to…
Following a problem posed by Lov\'asz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that…
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…
After long-term efforts, the Hamilton path (cycle) problem for connected vertex-transitive graphs of order $pq$ (where $p$ and $q$ are primes) was finally resolved in 2021, see [10]. Fifteen years ago, mathematicians began addressing this…
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…
We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbor model, there…
In this paper, the concept of cyclic subsets in graph theory is introduced. An interesting theorem which relates to the collective Hamiltonicity of these cyclic subsets in graphs is also presented. This paper uses this theorem to construct…
It was shown by Kutnar, Maru\v si\v c and Zhang in 2012 that every connected vertex-transitive graph of order $10p$, where $p$ is a prime and $p\ne 7$, contains a Hamilton path, except for graphs $X$ arising from the action of PSL$(2, s^m)$…
We show that there is an absolute constant $c>0$ such that every large connected $n$-vertex Cayley graph with degree $d\geq n^{1-c}$ has a Hamilton cycle. This makes progress towards the Lov\'asz conjecture and improves upon the previous…
In 1973, Chv\'atal conjectured that there exists a constant $t_0$ such that every $t_0$-tough graph on at least three vertices is Hamiltonian. While this conjecture is still open, work has been done to confirm it for several graph classes,…
One of the most well-known conjectures concerning Hamiltonicity in graphs asserts that any sufficiently large connected vertex transitive graph contains a Hamilton cycle. In this form, it was first written down by Thomassen in 1978,…
It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each…