A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a complete multipartite digraph. Such a digraph D is called ordinary if for any pair X, Y of its partite sets the set of arcs with end vertices in X ∪ Y coincides with X × Y = {x, y) : x ∈ X, y ∈ Y } or Y ×X or X×Y ∪Y ×X. We characterize a...

In [19] Huang gave a characterization of local tournaments. His characterization involves arc-reversals and therefore may not be easily used to solve other structural problems on locally semicomplete digraphs (where one deals with a fixed locally semicomplete digraph). In this paper we derive a classification of locally semicomplete digraphs which is very useful for studying structural properti...

Given integers k, s, t with 0 ≤ s ≤ t and k ≥ 0, a (k, t, s)-linear forest F is a graph that is the vertex disjoint union of t paths with a total of k edges and with s of the paths being single vertices. If the number of single vertex paths is not critical, the forest F will simply be called a (k, t)-linear forest. A graph G of order n ≥ k + t is (k, t)-hamiltonian if for any (k, t)-linear fore...

Abstract: A hypertournament or a k-tournament, on n vertices, 2≤k≤n, is a pair T= (V,E), where the vertex setV is a set of size n and the edge setE is the collection of all possible subsets of size k of V, called the edges, each taken in one of its k! possible permutations. A k-tournament is pancyclic if there exists (directed) cycles of all possible lengths; it is vertex-pancyclic if moreover ...

This paper generalises the concept of vertex pancyclic graphs. We define a graph as set-pancyclic if for every set S of vertices there is a cycle of every possible length containing S. We show that if the minimum degree of a graph exceeds half its order then the graph is set-pancyclic. We define a graph as k-ordered-pancyclic if, for every set S of cardinality k and every cyclic ordering of S, ...

For a pair of vertices u, v ∈ V (G), a cycle is called a geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle. A graph G is pancyclic [12] if it contains a cycle of every length from 3 to |V (G)| inclusive. Furthermore, a graph G is called geodesic k-pancyclic [3] if for each pair of vertices u, v ∈ V (G), it contains a geodesic cycle of every integer length of ...

For integers k and n with 2 ≤ k ≤ n− 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum d...

The Möbius cube MQn proposed by Cull et al. is an alternative to the popular hypercube network. Recently, MQn was shown to be pancyclic, i.e., cycles of any lengths at least four can be embedded into it. Due to the importance of the fault tolerance in the parallel processing area, in this paper, we study an injured MQn with mixed node and link faults. We show that it is (n − 2)-fault-tolerant p...

A property P defined on all graphs of order n is said to be k-stable if for any graph of order n that does not satisfy P , the fact that uv is not an edge of G and that G+uv satisfies P implies dG(u)+dG(v) < k. Every property is (2n−3)-stable and every k-stable property is (k+1)stable. We denote by s(P ) the smallest integer k such that P is k-stable and call it the stability of P . This number...

In this paper we consider uniquely pancyclic graphs, i.e. n vertex graphs with exactly one cycle of each length from 3 to n. The first result of the paper gives new upper and lower bounds on the number of edges in a uniquely pancyclic graph. Next we report on a computer search for new uniquely pancyclic graphs. We found that there are no new such graphs on n ≤ 59 vertices and that there are no ...