For simple graphs G1 and G2, the size Ramsey multipartite number mj(G1, G2) is defined as the smallest natural number s such that any arbitrary two coloring of the graph Kj×s using the colors red and blue, contains a red G1 or a blue G2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers mj(nK2, Cm) for j ≥ 2 and m ∈ {3, 4, 5, 6}.

We present entanglement witness operators for detecting genuine multipartite entanglement. These witnesses are robust against noise and require only two local measurement settings when used in an experiment, independent of the number of qubits. This allows detection of entanglement for an increasing number of parties without a corresponding increase in effort. The witnesses presented detect sta...

In this note, we investigate three versions of the overfull property for graphs and their relation to the edge-coloring problem. Each of these properties implies that the graph cannot be edge-colored with colors, where is the maximum degree. The three versions are not equivalent for general graphs. However, we show that some equivalences hold for the classes of indiierence graphs, split graphs,...

A groupie in a graph is a vertex whose degree is not less than the average degree of its neighbors. Under some mild conditions, we show that the proportion of groupies is very close to 1/2 in multitype random graphs (such as stochastic block models), which include Erdős-Rényi random graphs, random bipartite, and multipartite graphs as special examples. Numerical examples are provided to illustr...

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 ...

The pseudoachromatic number of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that, between any two distinct parts, there is at least one edge of G. This parameter is determined for graphs such as cycles, paths, wheels, certain complete multipartite graphs, and for other classes of graphs. Some open problems are raised.

A graph G has a representation modulo r if there exists an injective map f : V (G)→ {0, 1, ..., r − 1} such that vertices u and v are adjacent if and only if f(u) − f(v) is relatively prime to r. The representation number rep(G) is the smallest r such that G has a representation modulo r. Following earlier work on stars, we study representation numbers of complete bipartite graphs and more gene...

In this paper we give a method to associate a graph with an arbitrary density matrix referred to a standard orthonormal basis in the Hilbert space of a finite dimensional quantum system. We study the related issues like classification of pure and mixed states, Von-Neumann entropy, separability of multipartite quantum states and quantum operations in terms of the graphs associated with quantum s...

In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n ≥ 5 be an arbitrary positive odd integer; then, in any two-coloring of the edges of the complete 5-partite graph K((n − 1)/2, (n − 1)/2, (n − 1)/2, (n − 1)/2, 1) there is a monochromatic Cn, ∗2000 Mathematics Subject Classification: 05C55, 05C38. †Research supported in part by the Nat...

We present a continuous-variable CV Gaussian analog of cluster states, a new class of CV multipartite entangled states that can be generated from squeezing and quantum nondemolition coupling HI= XAXB. The entanglement properties of these states are studied in terms of classical communication and local operations. The graph states as general forms of the cluster states are presented. A chain for...