Related papers: Backward Arcs in Hamilton Oriented Cycles and Path…

A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a graph $G$ satisfies $\kappa(G) \ge \alpha(G)$, where $\kappa(G)$ is the vertex-connectivity of $G$, then $G$ has a Hamilton cycle. We prove a similar result…

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…

Let $G$ be a graph on $n$ vertices, $p$ the order of a longest path and $\kappa$ the connectivity of $G$. In 1989, Bauer, Broersma Li and Veldman proved that if $G$ is a 2-connected graph with $d(x)+d(y)+d(z)\ge n+\kappa$ for all triples…

A special case of a conjecture by Thomass\'e is that any oriented graph with minimum outdegree k contains a dipath of length 2k. For the sake of proving whether or not a counterexample exists, we present reductions and establish bounds on…

Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$…

A particular case of Caccetta-H\"{a}ggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order…

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…

A graph on $n$ vertices is called pancyclic if it contains a cycle of every length $3\le l \le n$. Given a Hamiltonian graph $G$ with independence number at most $k$ we are looking for the minimum number of vertices $f(k)$ that guarantees…

The dichromatic number $\vec{\chi}(G)$ of a digraph $G$ is the least integer $k$ such that $G$ can be partitioned into $k$ acyclic digraphs. A digraph is $k$-dicritical if $\vec{\chi}(G) = k$ and each proper subgraph $H$ of $G$ satisfies…

We say a graph $G$ has a Hamiltonian path if it has a path containing all vertices of $G$. For a graph $G$, let $\sigma_2(G)$ denote the minimum degree sum of two nonadjacent vertices of $G$; restrictions on $\sigma_2(G)$ are known as…

We study the existence of oriented paths with two blocks in oriented graphs under semidegree conditions. A block of an oriented path is a maximal directed subpath. Given positive integers $k$ and $\ell$ with $k/2\le \ell < k$, we establish…

The diameter of a directed graph is the maximum distance between any pair of vertices. We study a problem that generalizes \textsc{Oriented Diameter}: For a given directed graph and a positive integer $d$, what is the minimum number of arc…

The Path Avoiding Forbidden Pairs problem (PAFP) asks whether, in a directed graph $G$ with terminals $s,t$ and a set $\mathcal{F}$ of forbidden vertex pairs, there is an $s$-$t$ path that contains at most one endpoint from each forbidden…

The P\'osa--Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Koml\'os, S\'ark\"ozy and Szemer\'edi proved the conjecture for sufficiently…

We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least $n/2$, where $n$ denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs…

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and…

Kronk introduced the $l$-path hamiltonianicity of graphs in 1969. A graph is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. We have shown that if $P=uvz$ is a 2-path of a 2-connected,…

It was shown by Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2022 that every induced $2$-edge path in a vertex-transitive graph closes to an induced cycle. Similar results were obtained for 3-edge paths closing to cycles in…

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest…

An orientation $D$ of a graph $G=(V,E)$ is a digraph obtained from $G$ by replacing each edge by exactly one of the two possible arcs with the same end vertices. For each $v \in V(G)$, the indegree of $v$ in $D$, denoted by $d^-_D(v)$, is…