We prove that if is a subgraph of complete multipartite graph , then contains connected component satisfying . use this to every 3-coloring the edges monochromatic with at least edges. further show such coloring has circuit fraction This verifies conjecture Conlon and Tyomkyn. Moreover, for general we -coloring

An ecard of a graph G is a subgraph formed by deleting an edge. A da-ecard specifies the degree of the deleted edge along with the ecard. The degree associated edge reconstruction number of a graph G, dern(G), is the minimum number of da-ecards that uniquely determines G. The adversary degree associated edge reconstruction number of a graph G, adern(G), is the minimum number k such that every c...

Ohba conjectured that every graph G with |V (G)|6 2χ(G)+1 has its choice number equal its chromatic number. The on-line choice number of a graph is a variation of the choice number defined through a two person game, and is always at least as large as its choice number. Based on the result that for k > 3, the complete multipartite graph K2?(k−1),3 is not on-line k-choosable, the on-line version ...

Let G be a simple undirected graph with node set V (G) and edge set E(G). We call a subset F ⊆ E(G) independent if F is contained in the edge set of a complete multipartite (not necessarily induced) subgraph of G, F is dependent otherwise. In this paper we characterize the independents and the minimal dependents of G. We note that every minimal dependent of G has size two if and only if G is fa...

We report on the experimental violation of multipartite Bell inequalities by entangled states of trapped ions. First, we consider resource states for measurement-based quantum computation of between 3 and 7 ions and show that all strongly violate a Bell-type inequality for graph states, where the criterion for violation is a sufficiently high fidelity. Second, we analyze Greenberger-Horne-Zeili...

The monochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the ex...

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, γ(G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on every vertex simultaneously, no matter how the pebbles are ...

In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determi...

Quantum networks are natural scenarios for the communication of information among distributed parties, and the arena of promising schemes for distributed quantum computation. Measurement-based quantum computing is a prominent example of how quantum networking, embodied by the generation of a special class of multipartite states called cluster states, can be used to achieve a powerful paradigm f...

In this paper necessary and sufficient conditions are found for an edge-colored graph H to be the homomorphic image of a 2-factorization of a complete multipartite graph G in which each 2-factor of G has the same number of components as its corresponding color class in H. This result is used to completely solve the problem of finding hamilton decompositions of Ka,b−E(U) for any 2-factor U of Ka...