We develop protocols for preparing a GHZ state and, in general, a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite ent...

Let [n]r be the complete r-partite hypergraph with vertex classes of size n. It is an easy exercise to show that every set of more than (k − 1)nr−1 edges in [n]r contains a matching of size k. We conjecture the following rainbow version of this observation: If F1, F2, . . . , Fk ⊆ [n]r are of size larger than (k− 1)nr−1 then there exists a rainbow matching, i.e. a choice of disjoint edges fi ∈ ...

We consider the problem of covering the complete r-uniform hypergraphs on n vertices using complete r-partite graphs. We obtain lower bounds on the size of such a covering. For small values of r our result implies a lower bound of Ω( e r r √ r n log n) on the size of any such covering. This improves the previous bound of Ω(rn log n) due to Snir [5]. We also obtain good lower bounds on the size ...

For every real p > 0 and simple graph G, set f (p,G) = ∑ u∈V (G) d (u) , and let φ (r, p, n) be the maximum of f (p,G) taken over all Kr+1-free graphs G of order n. We prove that, if 0 < p < r, then φ (r, p, n) = f (p, Tr (n)) , where Tr (n) is the r-partite Turan graph of order n. For every p ≥ r + ⌈√ 2r ⌉ and n large, we show that φ (p, n, r) > (1 + ε) f (p, Tr (n)) for some ε = ε (r) > 0. Ou...

A rate R = 5/20 hypergraph-based woven convolutional code with overall constraint length 67 and constituent convolutional codes is presented. It is based on a 3-partite, 3uniform, 4-regular hypergraph and contains rate R = 3/4 constituent convolutional codes with overall constraint length 5. Although the code construction is based on low-complexity codes, the free distance of this construction,...

A famous conjecture (usually called Ryser’s conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an r-uniform r-partite hypergraph H, the inequality τ(H) 6 (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem, and in the case of r = 3, which was proved by Aharoni in 2001. Here we ...

Let Fn,tr(n) denote the family of all graphs on n vertices and tr(n) edges, where tr(n) is the number of edges in the Turán’s graph Tr(n) – the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G with at most λ colors, and let f(n, tr(n), λ) = max{PG(λ) : G ∈ Fn,tr(n...

We call a graph G O-graph if there is an optimal coloring of the set of vertices and an optimal (disjoint) covering with cliques such that any class of colors intersects any clique. In this paper, it has been established the relation to [p, q, r]-partite graphs and the fact that the O-graphs admit a k-clique-join.

For a fixed r, let fr(n) denote the minimum number of complete r-partite rgraphs needed to partition the complete r-graph on n vertices. The Graham-Pollak theorem asserts that f2(n) = n − 1. An easy construction shows that fr(n) 6 (1 + o(1)) ( n br/2c ) , and we write cr for the least number such that fr(n) 6 cr(1 + o(1)) ( n br/2c ) . It was known that cr < 1 for each even r > 4, but this was ...

Graphs with large minimum degree containing no copy of a clique on r vertices (Kr) must contain relatively large independent sets. A classical result of Andrásfai, Erdős, and Sós implies that Kr-free graphs G with degree larger than ((3r−7)/(3r− 4))|V (G)| must be (r− 1)-partite. An obvious consequence of this result is that the same degree threshold implies an independent set of order (1/(r − ...