The notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi–coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition...

A graph G is r-equitably k-colorable if its vertex set can be partitioned into k independent sets, any two of which differ in size by at most r. The r-equitable chromatic threshold of a graph G, denoted by χr=(G), is the minimum k such that G is r-equitably k -colorable for all k ≥ k. Let G × H denote the Kronecker product of graphs G and H. In this paper, we completely determine the exact valu...

The choice number of a graph G is the smallest integer k such that for every assignment of a list L(v) of k colors to each vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from L(v). We present upper and lower bounds on the choice number of complete multipartite graphs with partite classes of equal sizes and complete r-partite graphs with r−1 partite classes ...

In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. 2 ) ( = G β ) is characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever 2 ) (...

Let G,J n) be an r-chromatic graph with n vertices in each colour class . Suppose G = G 3 (n), and t (G) . the minimal degree in G, is at least n + t (t _> 1) . We prove that C contains at least t 3 triangles but does not have to contain more titan 4t 3 of them . Furthermore, we give lower bounds for s such that G contains a complete 3-partite graph with s vertices in each class . Let';.(ii) = ...

For n > r > 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete r-graph on n vertices into q complete r-partite r-graphs. Graham and Pollak showed that fz(n) = n-1. Here we observe that f3(n) = n-2 and show that for every fixed r > 2, there are positive constants cx(r) and c2(r) such that q(r) < f,(n)" n-f'/2J < c2(r) for all n > r. This sol...

Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H , each satisfying the following condition: the degree of each legal r−1-tuple contained in the complement of this side is strictly larger than n 2 . We prove that under this condition H must have a perfect matching. This answers a question of Kühn and Osthus.

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...

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 $[k,n,1]$-graph is a $k$-partite graph with parts of order $n$ such that the bipartite induced by any pair matching. An independent transversal in an set intersects each part single vertex. factor transversals pairwise-disjoint transversals. Let $f(k)$ be smallest integer $n_0$ every has assuming $n \ge n_0$. Several known conjectures imply for $k 2$, $f(k)=k$ if $k$ even and $f(k)=k+1$ odd. ...