Perfect 1-factorisations of circulants with small degree

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed

On the perfect 1-factorisation problem for circulant graphs of degree 4

A 1-factorisation of a graph G is a partition of the edge set of G into 1 factors (perfect matchings); a perfect 1-factorisation of G is a 1-factorisation of G in which the union of any two of the 1-factors is a Hamilton cycle in G. It is known that for bipartite 4-regular circulant graphs, having order 2 (mod 4) is a necessary (but not sufficient) condition for the existence of a perfect 1-factorisation. The only known non-bipartite 4-regular circulant graphs that admit a perfect 1-factorisation are trivial (on 6 vertices). We prove several construction results for perfect 1-factorisations of a large class of bipartite 4-regular circulant graphs. In addition, we show that no member of an infinite family of non-bipartite 4-regular circulant graphs admits a perfect 1-factorisation. This supports the conjecture that there are no perfect 1-factorisations of any connected non-bipartite 4-regular circulant graphs of order at least 8

Graph Theoretic Modeling: Case Studies In Redundant Arrays Of Independent Disks And Network Defense

Graph theoretic modeling has served as an invaluable tool for solving a variety of problems since its introduction in Euler\u27s paper on the Bridges of Königsberg in 1736 . Two amongst them of contemporary interest are the modeling of Redundant Arrays of Inexpensive Disks (RAID), and the identification of network attacks. While the former is vital to the protection and uninterrupted availability of data, the latter is crucial to the integrity of systems comprising networks. Both are of practical importance due to the continuing growth of data and its demand at increasing numbers of geographically distributed locations through the use of networks such as the Internet. The popularity of RAID has soared because of the enhanced I/O bandwidths and large capacities they offer at low cost. However, the demand for bigger capacities has led to the use of larger arrays with increased probability of random disk failures. This has motivated the need for RAID systems to tolerate two or more disk failures, without sacrificing performance or storage space. To this end, we shall first perform a comparative study of the existing techniques that achieve this objective. Next, we shall devise novel graph-theoretic algorithms for placing data and parity in arrays of n disks (n ≥ 3) that can recover from two random disk failures, for n = p - 1, n = p and n = 2p - 2, where p is a prime number. Each shall be shown to utilize an optimal ratio of space for storing parity. We shall also show how to extend the algorithms to arrays with an arbitrary number of disks, albeit with non-optimal values for the aforementioned ratio. The growth of the Internet has led to the increased proliferation of malignant applications seeking to breach the security of networked systems. Hence, considerable effort has been focused on detecting and predicting the attacks they perpetrate. However, the enormity of the Internet poses a challenge to representing and analyzing them by using scalable models. Furthermore, forecasting the systems that they are likely to exploit in the future is difficult due to the unavailability of complete information on network vulnerabilities. We shall present a technique that identifies attacks on large networks using a scalable model, while filtering for false positives and negatives. Furthermore, it also forecasts the propagation of security failures proliferated by attacks over time and their likely targets in the future

A family of perfect factorisations of complete bipartite graphs

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n = p(2) for an odd prime p. We construct a family of (p-1)/2 non-isomorphic perfect 1-factorisations of K-n,K-n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltoilian if the permutation defined by any row relative to any other row is a single Cycle. (C) 2002 Elsevier Science (USA)