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.

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.

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)}.

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.

Packing complete multipartite graphs with 4-cycles

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge-disjoint 4-cycles, and the minimum leaves are explicitly given.

Decomposition of complete tripartite graphs into gregarious 4-cycles

A 4-cycle in a tripartite graph with vertex partition {V-1, V-2, V-3} is said to be gregarious if it has at least one vertex in each V-i, 1 less than or equal to i less than or equal to 3. In this paper, necessary and sufficient conditions are given for the existence of an edge-disjoint decomposition of any complete tripartite graph into gregarious 4-cycles.

Decomposing complete tripartite graphs into cycles of lengths 3 and 4

Necessary and sufficient conditions are given for the edge-disjoint decomposition of a complete tripartite graph K-r,K-s,K-t into exactly alpha 3-cycles and beta 4-cycles. (C) 1999 Elsevier Science B.V. All rights reserved.

Late Quaternary cycles of mangrove development and decline on the north Australian continental shelf

Mangrove communities in the Australian tropics presently occur as narrow belts of vegetation in estuaries and on sheltered, muddy coasts. Palynological data from continental shelf and deep-sea cores indicate a long-term cyclical component of mangrove development and decline at a regional scale, which can be linked to specific phases of late Quaternary sealevel change. Extensive mangrove development, relative to today, occurs during periods of marine transgression, whereas very diminished mangrove occurs during marine regressions and during rarer periods of relative sea-level stability. Episodes of flourishing mangrove cannot be linked to phases of humid climate, as has been suggested in studies elsewhere. Rather, the cycle of expansion and decline of mangrove communities on a grand scale is explained in terms of contrasting physiographic settings characteristic of continental-shelf coasts during transgressive and regressive phases, in particular by the existence, or lack, of well-developed tidal estuaries. Copyright (C) 1999 John Wiley & Sons, Ltd.

Implication of the visual system in the regulation of activity cycles in the absence of solar light: 2-[I-125]iodomelatonin binding sites and melatonin receptor gene expression in the brains of demersal deep-sea gadiform fish

Relative eye size, gross brain morphology and central localization of 2-[I-125]iodomelatonin binding sites and melatonin receptor gene expression were compared in six gadiform fish living at different depths in the north-east Atlantic Ocean: Phycis blennoides (capture depth range 265-1260 m), Nezumia aequalis (445-1512 m), Coryphaenoides rupestris (706-1932 m), Trachyrincus murrayi (1010-1884 m), Coryphaenoides guentheri (1030 m) and Coryphaenoides (Nematonurus) armatus (2172-4787 m). Amongst these, the eye size range was 0.15-0.35 of head length with a value of 0.19 for C.(N.) armatus, the deepest species. Brain morphology reflected behavioural differences with well-developed olfactory regions in P.blennoides, T.murrayi and C. (N.) armatus and evidence of olfactory deficit in N. aequalis, C. rupestris and C. guentheri. All species had a clearly defined optic tectum with 2-[I-125] iodomelatonin binding and melatonin receptor gene expression localized to specific brain regions in a similar pattern to that found in shallow-water fish. Melatonin receptors were found throughout the visual structures of the brains of all species. Despite living beyond the depth of penetration of solar light these fish have retained central features associated with the coupling of cycles of growth, behaviour and reproduction to the diel light-dark cycle. How this functions in the deep sea remains enigmatic.

The number of 4-cycles in 2-factorizations of K2n minus a 1-factor

In this paper we give a complete solution to problem of determining the number of 4-cycles in a 2-factorization of K-2n\ 1-factor. (C) 2000 Elsevier Science B.V. All rights reserved.

Decompositions of complete multipartite graphs into cycles of even length

Necessary and sufficient conditions for the existence of an edge-disjoint decomposition of any complete multipartite graph into even length cycles are investigated. Necessary conditions are listed and sufficiency is shown for the cases when the cycle length is 4, 6 or 8. Further results concerning sufficiency, provided certain small decompositions exist, are also given for arbitrary even cycle lengths.

On the Oberwolfach problem with two similar length cycles

For all in greater than or equal to 3, the Oberwolfach problem is solved for the case where the 2-factors consist of two cycles of lengths in and m + 1, and for the case where the 2-factors consist of two cycles of lengths m and m + 2.

Japanese encephalitis on Badu Island, Australia: the first isolation of Japanese encephalitis virus from Culex gelidus in the Australasian region and the role of mosquito host-feeding patterns in virus transmission cycles

During investigation of an outbreak of Japanese encephalitis (JE) in the Torres Strait, Australia, in 2000, mosquitoes were collected in Badu Island community and at a newly established communal piggery about 3 km from the community. A total of 94285 mosquitoes, comprising 91240 (96.8%) unengorged females, 1630 (1.7%) blood-engorged females and 1415 (1.5%) males, were processed for virus isolation. One isolate of JE virus was obtained from Culex gelidus, with a minimum infection rate of 12.4:1000. This is the first isolate of JE virus from Cx. gelidus in the Australasian region. No isolates were obtained from Cx. annulirostris, the primary implicated Australian JE vector. Analysis of mosquito host-feeding patterns, using gel diffusion, demonstrated that Cx. annulirostris and 5 other species fed predominately on mammals, Analysis of blood-fed mosquitoes collected within the community demonstrated that the proportion of Cx. annulirostris feeding on pigs in 2000 (2.3%) was significantly lower than that for the 1995-97 period (31.3%). The removal of the pigs from Badu Island community has limited the contact between potential amplifying hosts and mosquitoes, thus potentially reducing the risk of transmission of JE virus to the human population.