Related papers: A sufficient condition for pre-Hamiltonian cycles …

Let $G$ be a graph. A total dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $G$ is adjacent to a vertex in $S$. Recently, the following question was proposed: "Is it true that every connected cubic…

A digraph $D=(V(D)$, $A(D))$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in \{3,\ldots,n\}$; and $D$ is vertex-pancyclic iff, for each vertex $v\in V(D)$ and each $k\in \{3,\ldots,n\}$,…

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…

Let $D$ be a strong digraph on $n=2m+1\geq 5$ vertices. In this paper we show that if $D$ contains a cycle of length $n-1$, then $D$ has also a cycle which contains all vertices with in-degree and out-degree at least $m$ (unless some…

In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least $k$, then it contains a cycle of…

We show that an edge-dominating cycle in a $2K_2$-free graph can be found in polynomial time; this implies that every 1/(k-1)-tough $2K_2$-free graph admits a k-walk, and it can be found in polynomial time. For this class of graphs, this…

An oriented graph is a digraph that contains no 2-cycles, i.e., there is at most one arc between any two vertices. We show that every oriented graph $G$ of sufficiently large order $n$ with $\mathrm{deg}^+(x) +\mathrm{deg}^{-}(y)\geq…

A hypergraph $\mathcal H$ is super-pancyclic if for each $A \subseteq V(\mathcal H)$ with $|A| \geq 3$, $\mathcal H$ contains a Berge cycle with base vertex set $A$. We present two natural necessary conditions for a hypergraph to be…

A covering of a digraph $D$ by Hamilton cycles is a collection of directed Hamilton cycles (not necessarily edge-disjoint) that together cover all the edges of $D$. We prove that for $1/2 \geq p\geq \frac{\log^{20} n}{n}$, the random…

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct…

Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for…

Suppose $\Gamma$ is a finite simple graph. If $D$ is a dominating set of $\Gamma$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $\Gamma$, then we say that $D$ is an odd dominating set for $\Gamma$. For a…

In [{Structural properties and decomposition of linear balanced matrices}, {\it Mathematical Programming}, 55:129--168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of…

We prove: (i) if $G$ is a 1-tough graph of order $n$ and minimum degree $\delta$ with $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle unless $G$ belongs to an easily specified class of graphs with $\kappa(G)=2$ and…

Let $G$ be a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$. It is proved that if $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.

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…

For fixed $k\ge 2$, determining the order of magnitude of the number of edges in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length $2k$, is a long-standing open problem. We consider an extension of this problem to…

A set of vertices $X\subseteq V(G)$ is a $d$-distance dominating set if for every $u\in V(G)\setminus X$ there exists $x\in X$ such that $d(u,x) \le d$, and $X$ is a $p$-packing if $d(u,v) \ge p+1$ for every different $u,v\in X$. The…

The dicycle transversal number t(D) of a digraph D is the minimum size of a dicycle transversal of D, i. e. a set T of vertices of D such that D-T is acyclic. We study the following problem: Given a digraph D, decide if there is a dicycle B…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…