For r 4 we determine the smallest number of vertices, .<71-( d), of complete that are decomposable into two isomorphic factors for a given finite diameter d. We also prove that for a ,d such graph exists for each order than gr( d). 1. INTRODUCTORY NOTES AND DEFINITIONS In this paper we study decompositions of finite complete multipartite graphs factors with prescribed diameter. A factor F of a ...

Entanglement bits or “ebits” have been proposed as a quantitative measure of a fundamental resource in quantum information processing. For such an interpretation to be valid, it is important to show that the same number of ebits in different forms or concentrations are inter-convertible in the asymptotic limit. Here we draw attention to a very important but hitherto unnoticed aspect of entangle...

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...

We strengthen the stability theorem of Erdős and Simonovits. Write Kr (s1, . . . , sr) for the complete r-partite graph with parts of size s1, . . . , sr and Tr (n) for the r-partite Turán graph of order n. Our main result is: For all r ≥ 2 and sufficiently small c > 0, ε > 0, for every graph G of order n with ⌈ (1− 1/r − ε)n2/2 ⌉ edges, one of the following conditions holds: (a) Kr+1 ( ⌊c lnn⌋...