Abstract: Random Intersection Graphs is a new class of random graphs introduced in [5], in which each of n vertices randomly and independently chooses some elements from a universal set, of cardinality m. Each element is chosen with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=n{\'a}, for any real {\'a} different than one, we establish here, for the first time, tight lower bounds p0(n,m), on the value of p, as a function of n and m, above which the graph Gn,m,p is almost certainly Hamiltonian, i.e. it contains a Hamilton Cycle almost certainly. Our bounds are tight in the sense that when p is asymptotically smaller than p0(n,m) then Gn,m,p almost surely has a vertex of degree less than 2. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection model. Interestingly, Hamiltonicity appears well below the general thresholds, of [4], at which Gn,m,p looks like a usual random graph. Thus our bounds are much stronger than the trivial bounds implied by those thresholds.
Our results strongly support the existence of a threshold for Hamiltonicity in Gn,m,p.

Abstract: We investigate the practical merits of a parallel priority queue
through its use in the development of a fast and work-efficient parallel
shortest path algorithm, originally designed for an EREW PRAM. Our
study reveals that an efficient implementation on a real supercomputer
requires considerable effort to reduce the communication performance
(which in theory is assumed to take constant time). It turns out that the
most crucial part of the implementation is the mapping of the logical
processors to the physical processing nodes of the supercomputer. We
achieve the requested efficient mapping through a new graph-theoretic
result of independent interest: computing a Hamiltoniancycle on a directed
hyper-torus. No such algorithm was known before for the case of
directed hypertori. Our Hamiltoniancycle algorithm allows us to considerably
improve the communication cost and thus the overall performance
of our implementation.