Sharp minimum degree and degree sum conditions are proven for the existence of a Hamiltonian cycle passing through specified vertices with prescribed distances between them in large graphs.

We say that a k-uniform hypergraph C is an `-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ` vertices. We prove that if 1 ≤ ` < k and k − ` does not divide k then any k-uniform hypergraph on n vertices with minim...

Moon [J. Combin. Inform. System Sci. 19 (1994), 207–214] showed that every strong tournament contains a Hamiltonian cycle through at least three pancyclic arcs. In this paper, we extend the result of Moon and prove that if D is a strong c-partite tournament with c ≥ 3, then D contains a cycle C containing vertices from exactly c partite sets such that C contains at least three arcs, each of whi...

We say that a k-uniform hypergraph C is a Hamilton cycle of type `, for some 1 ≤ ` ≤ 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 of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have |Ei−1 \ Ei| = `. We define a class of ( , p)-regular hypergraphs, that includes random hypergraph...

In a biconnected graph, a detour is the best alternative path from a detour-starting vertex to the destination vertex. A detour-starting vertex is the vertex from which the original shortest path is changed. The longest detour (LD) problem is to find a detour-critical edge in a shortest path such that the removal of the edge results in the maximum distance increment. In this paper, we deal with...

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2-connected weighted graph such that the minimum weighted degree of G is at lea...

The vertex cover problem and the vertex partition problem are central problems in graph theory and many generalizations are known. Two examples are the minimum k-path vertex cover problem (MkPVCP for short, introduced in [1]), which asks for a minimum vertex sets covering every path of length k−1, and the minimum k-path partition problem (MkPPP for short, introduced in [2]), which asks for a mi...

A vertex set S ⊆ V (G) is k-weak-edge-connected if for every C ⊂ S and x ∈ S−C there are min{k, |C|} edge-disjoint (x,C)-paths in G. For a graph G and a k-weakedge-connected vertex set S ⊂ V (G) with k ≥ 3 and 4 ≤ |S| ≤ 2k, we show that G has an eulerian subgraph containing all vertices in S.

Assume that n is a positive integer with n 2. It is proved that between any two different vertices x and y of Qn there exists a path Pl(x,y) of length l for any l with h(x,y) l 2n − 1 and 2|(l − h(x,y)). We expect such path Pl(x,y) can be further extended by including the vertices not in Pl(x,y) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove t...