A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d + (x) be the outdgree and d ? (x) the indegree of the vertex x in D. The minimum (maximum) out-degree and the minimum (maximum) indegree of D are denoted by + ((+) and ? ((?), respectively. In addition, we deene = minf + ; ?...

For positive integers $s,t,u,v$, we define a bipartite graph $\Gamma_{\mathbb{R}}(X^s Y^t,X^u Y^v)$ where each partite set is copy of $\mathbb{R}^3$, and vertex $(a_1,a_2,a_3)$ in the first adjacent to $[x_1,x_2,x_3]$ second if only \[ a_2 + x_2 = a_1^s x_1^t \quad \text{and} a_3+x_3=a_1^ux_1^v. \] In this paper, classify all such graphs according girth.

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...

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem only known to be true $r\leq 3$, or when $\nu(H)=1$ and 5$. An equivalent formulation in $r$-edge coloring a graph $G$ independence $\alpha(G)$, there exists $(r-1)\alpha(G)$ monochromatic connected subgra...

A sequence of nonnegative integers π is graphic if it is the degree sequence of some graph G. In this case we say that G is a realization of π, and we write π = π(G). A graphic sequence π is potentially H-graphic if there is a realization of π that contains H as a subgraph. Given nonincreasing graphic sequences π1 = (d1, . . . , dn) and π2 = (s1, . . . , sn), we say that π1 majorizes π2 if di ≥...

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 ...