Hughie Hamilton, Alex McDonald during Ky visit demonstration

Group of people including Hughie Hamilton, Alex McDonald and Vilma Ward during visit of former South Vietnamese vice president Nguyen Cao Ky to Brisbane, Australia in January 1967.

Maximal sets of Hamilton cycles in Kn,n

In this article, we prove that there exists a maximal set of m Hamilton cycles in K-n,K-n if and only if n/4 < m less than or equal to n/2. (C) 2000 John Wiley & Sons, Inc.

On the Hamilton-Waterloo problem

The Hamilton-Waterloo problem asks for a 2-factorisation of K-v in which r of the 2-factors consist of cycles of lengths a(1), a(2),..., a(1) and the remaining s 2-factors consist of cycles of lengths b(1), b(2),..., b(u) (where necessarily Sigma(i)(=1)(t) a(i) = Sigma(j)(=1)(u) b(j) = v). In thus paper we consider the Hamilton-Waterloo problem in the case a(i) = m, 1 less than or equal to i less than or equal to t and b(j) = n, 1 less than or equal to j less than or equal to u. We obtain some general constructions, and apply these to obtain results for (m, n) is an element of {(4, 6)1(4, 8), (4, 16), (8, 16), (3, 5), (3, 15), (5, 15)}.

Decompositions of complete graphs into triangles and Hamilton cycles

For all odd integers n greater than or equal to 1, let G(n) denote the complete graph of order n, and for all even integers n greater than or equal to 2 let G,, denote the complete graph of order n with the edges of a 1-factor removed. It is shown that for all non-negative integers h and t and all positive integers n, G, can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G(n). (C) 2004 Wiley Periodicals, Inc.

Visualisation of macropinosome maturation by the recruitment of sorting nexins

We report that phosphoinositol-binding sorting nexin 5 ( SNX5) associates with newly formed macropinosomes induced by EGF stimulation. We used the recruitment of GFP-SNX5 to macropinosomes to track their maturation. Initially, GFP-SNX5 is sequestered to discrete subdomains of the macropinosome; these subdomains are subsequently incorporated into highly dynamic, often branched, tubular structures. Time-lapse videomicroscopy revealed the highly dynamic extension of SNX5-labelled tubules and their departure from the macropinosome body to follow predefined paths towards the perinuclear region of the cell, before fusing with early endosomal acceptor membranes. The extension and departure of these tubular structures occurs rapidly over 5-10 minutes and is dependent upon intact microtubules. As the tubular structures depart from the macropinosome there is a reduction in the surface area and an increase in tension of the limiting membrane of the macropinosome. In addition to the recruitment of SNX5 to the macropinosome, Rab5, SNX1 and EEA1 are also recruited by newly formed macropinosomes, followed by the accumulation of Rab7. SNX5 forms heterodimers with SNX1 and this interaction is required for endosome association of SNX5. We propose that the departure of SNX5-positive tubules represents a rapid mechanism of recycling components from macropinosomes thereby promoting their maturation into Rab7-positive structures. Collectively these findings provide a detailed real-time characterisation of the maturation process of the macropinocytic endosome.

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.