In the following material, we use the notion of an independent set of a graph G: this is a set of I vertices of G such that no two vertices of I are adjacent in G. These are sometimes referred to as stable sets in the literature. The chromatic number of a graph G, denoted χ(G), is the smallest r such that V (G) has a partition into independent sets V1, V2, . . . , Vr. In other words, we may ass...

In this thesis, we study the minimum vertex cover problem on the class of k-partite kuniform hypergraphs. This problem arises when reducing the size of nondeterministic finite automata (NFA) using preorders, as suggested by Champarnaud and Coulon. It has been shown that reducing NFAs using preorders is at least as hard as computing a minimal vertex cover on 3-partite 3-uniform hypergraphs, whic...

We present a general framework for studying multi-particle entanglement using graph-theoretic methods. We introduce a technique called bicolored merging for determining combinatorial conditions that must be satisfied for two multiparticle states to be comparable under LOCC. We present several results concerning the possibility and impossibility of various state transformations of multi-partite ...

A digraph obtained by replacing each edge of a complete mpartite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete m-partite digraph. We describe an O(n) algorithm for finding a longest path in a complete m-partite (m ≥ 2) digraph with n vertices. The algorithm requires time O(n) in case of testing only the existence of a Hamiltonian path and ...

We show that if r s 2; n > r8; and G is a graph of order n containing as many r-cliques as the r-partite Turán graph of order n; then G has more than nr 1=r2r+12 cliques sharing a common edge unless G is isomorphic to the r-partite Turán graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.

The equivalence problem under local unitary transformation for n–partite pure states is reduced to the one for (n− 1)–partite mixed states. In particular, a tripartite system HA⊗HB ⊗HC , where Hj is a finite dimensional complex Hilbert space for j = A,B,C, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace...

Let r ≥ 2 and write K+ r (s1, . . . , sr) for the complete r-partite graph with parts of size s1 ≥ 2, . . . , sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove the following theorems: (A) For all r ≥ 2, sufficiently small ε > 0 and sufficiently large n, every graph of order n with tr (n) + 1 edges contains a K + r ( ...

We show that one single-mode squeezed state distributed among N parties using linear optics suffices to produce a truly N-partite entangled state for any nonzero squeezing and arbitrarily many parties. From this N-partite entangled state, via quadrature measurements of N-2 modes, bipartite entanglement between any two of the N parties can be "distilled," which enables quantum teleportation with...