An n-tournament is an orientation of a complete n-partite graph. It was proved by J.A. Bondy in 1976 that every strongly connected n-partite tournament has an n-cycle. We characterize strongly connected n-partite tournaments in which a longest cycle is of length n and, thus, settle a problem in L. Volkmann, Discrete Math. 245 (2002) 19-53.

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...

Quantifying genuine entanglement is a crucial task in quantum information theory.In this work, we give an approach of constituting $m$-partite measures from any bipartite and $k$-partite measure, $3\leq k<m$. In addition, as complement to the three-qubit concurrence triangle proposed [Phys. Rev. Lett., 127, 040403], show that relation also valid for continuous measure system with dimension. We ...

If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is almost regular. A c-partite tournament is an orientation of a complete c-...

A digraph D is called is semicomplete c-partite if its vertex set V (D) can be partitioned into c sets (partite sets) such that for any two vertices x and y in diierent partite sets at least one arc between x and y is in D and there are no arcs between vertices in the same partite set. The path covering number of D is the minimum number of paths in D that are pairwise vertex disjoint and cover ...

A multipartite or c-partite tournament is an orientation of a complete c-partite graph. Lu and Guo (submitted for publication) [3] recently introduced strong quasi-Hamiltonianconnectivity of a multipartite tournament D as follows: For any two distinct vertices x and y ofD, there is a pathwith at least one vertex from each partite set ofD from x to y and from y to x. We obtain the definition for...

This paper provides lower orientable k-step domination number and upper orientable k-step domination number of complete r-partite graph for 1 ≤ k ≤ 2. It also proves that the intermediate value theorem holds for the complete r-partite graphs.

In [8], it is shown that the complete multipartite graph Kn(2t) having n partite sets of size 2t, where n ≥ 6 and t ≥ 1, has a decomposition into gregarious 6-cycles if n ≡ 0, 1, 3 or 4 (mod 6). Here, a cycle is called gregarious if it has at most one vertex from any particular partite set. In this paper, when n ≡ 0 or 3 (mod 6), another method using difference set is presented. Furthermore, wh...

An [n, k, r]-partite graph is a graph whose vertex set, V , can be partitioned into n pairwisedisjoint independent sets, V1, . . . , Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 ≤ i < j ≤ n. An independent transversal in an [n, k, r]-partite graph is an independent set, T , consisting of n vertices, one from each Vi. An ...