A note on the Balanced method

Recently the Balanced method was introduced as a class of quasi-implicit methods for solving stiff stochastic differential equations. We examine asymptotic and mean-square stability for several implementations of the Balanced method and give a generalized result for the mean-square stability region of any Balanced method. We also investigate the optimal implementation of the Balanced method with respect to strong convergence.

Decomposition of complete graphs into 5-cubes

Necessary conditions for the complete graph on n vertices to have a decomposition into 5-cubes are that 5 divides it - 1 and 80 divides it (it - 1)/2. These are known to be sufficient when n is odd. We prove them also sufficient for it even, thus completing the spectrum problem for the 5-cube and lending further weight to a long-standing conjecture of Kotzig. (c) 2005 Wiley Periodicals, Inc.

Some equitably 2-colourable cycle decompositions of multipartite graphs

Let G be a graph in which each vertex has been coloured using one of k colours, say c(1), c(2),..., c(k). If an m-cycle C in G has x(i) vertices coloured c(i), i = 1, 2,..., k, and vertical bar x(i) - x(j)vertical bar

On Hamilton cycle decomposition of 6-regular circulant graphs

The circulant graph Sn, where S ⊆ Zn \ {0}, has vertex set Zn and edge set {{x, x + s}|x ∈ Zn, s ∈ S}. It is shown that there is a Hamilton cycle decomposition of every 6-regular circulant graph Sn in which S has an element of order n.