Years and Authors of Summarized Original Work
2005; Efthymiou, Spirakis
Problem Definition
E. Marczewski proved that every graph can be represented by a list of sets where each vertex corresponds to a set and the edges to nonempty intersections of sets. It is natural to ask what sort of graphs would be most likely to arise if the list of sets is generated randomly.
Consider the model of random graphs where each vertex chooses randomly from a universal set the members of its corresponding set, each independently of the others. The probability space that is created is the space of random intersection graphs, \( { G_{n,m,p} } \), where n is the number of vertices, m is the cardinality of a universal set of elements and p is the probability for each vertex to choose an element of the universal set. The model of random intersection graphs was first introduced by M. Karońsky, E. Scheinerman, and K. Singer-Cohen in [4]. A rigorous definition of the model of random intersection graphs follows:
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Efthymiou, C., Spirakis, P.(. (2016). Hamilton Cycles in Random Intersection Graphs. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_176
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