Let T be a 3-partite tournament. We say that a vertex v is −→ C3 -free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3 -free vertices in a 3-partite tournament and f3(T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F3(T )must be contained in one of the partite sets of T . It is also shown that for every regular 3...

A signed k-partite graph (signed multipartite graph) is a k-partite graph in which each edge is assigned a positive or a negative sign. If G(V1, V2, · · · , Vk) is a signed k-partite graph with Vi = {vi1, vi2, · · · , vini}, 1 ≤ i ≤ k, the signed degree of vij is sdeg(vij) = dij = d + ij − d − ij , where 1 ≤ i ≤ k, 1 ≤ j ≤ ni and d + ij(d − ij) is the number of positive (negative) edges inciden...

We give a new separability criterion, a necessary condition for separability of N partite quantum states. The criterion is based on the Bloch representation of a N -partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N -partite quantum states in H = H1 ⊗H2 ⊗ · · ·⊗HN . The criterion can test whether a N -parti...

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call (D)=max16 i6 n{|Vi|} the independence number of the semicomplete n-partite digraph D, where V1; V2; : : : ; Vn are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and t...

The complete multipartite graph Kn(2t) having n partite sets of size 2t, with n ≥ 6 and t ≥ 1, is shown to have a decomposition into gregarious 6-cycles, that is, the cycles which have at most one vertex from any particular partite set. Complete sets of differences of numbers in Zn are used to produce starter cycles and obtain other cycles by rotating the cycles around the n-gon of the partite ...

Let T be a 3-partite tournament. We say that a vertex v is −→ C3-free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3-free vertices in a 3-partite tournament and f3(T ) its cardinality. In a recent paper, it was proved that if T is a regular 3-partite tournament, then f3(T ) < n 9 , where n is the order of T . In this paper, we prove that f3(T ) ≤ n 12 . 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-...

An orientation of a complete graph is a tournament, and an orientation of a complete n-partite graph is an n-partite tournament. For each n 2:: 4, there exist examples of strongly connected n-partite tournament without any strongly connected subtournaments of order p 2:: 4. If D is a digraph, then let d+ (x) be the out degree and d(x) the indegree of the vertex x in D. The minimum (maximum) out...

Koh and Tan gave a sufficient condition for a 3-partite tournament to have at least one 3-king in [K.M. Koh, B.P. Tan, Kings in multipartite tournaments, Discrete Math. 147 (1995) 171–183, Theorem 2]. In Theorem 1 of this paper, we extend this result to n-partite tournaments, where n 3. In [K.M. Koh, B.P. Tan, Number of 4-kings in bipartite tournaments with no 3-kings, Discrete Math. 154 (1996)...