Permutation Pseudographs and Contiguity

  • Authors:

  • Catherine Greenhill;Svante Janson;Jeong Han Kim;Nicholas C. Wormald

  • Affiliations:

  • Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia (email: {csg,nick}@ms.unimelb.edu.au);Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden (email: svante@math.uu.se);Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (email: jehkim@microsoft.com);Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia (email: {csg,nick}@ms.unimelb.edu.au)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2002

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Abstract

The space of permutation pseudographs is a probabilistic model of 2-regular pseudographs on n vertices, where a pseudograph is produced by choosing a permutation &sgr; of {1,2,…, n} uniformly at random and taking the n edges {i,&sgr;(i)}. We prove several contiguity results involving permutation pseudographs (contiguity is a kind of asymptotic equivalence of sequences of probability spaces). Namely, we show that a random 4-regular pseudograph is contiguous with the sum of two permutation pseudographs, the sum of a permutation pseudograph and a random Hamilton cycle, and the sum of a permutation pseudograph and a random 2-regular pseudograph. (The sum of two random pseudograph spaces is defined by choosing a pseudograph from each space independently and taking the union of the edges of the two pseudographs.) All these results are proved simultaneously by working in a general setting, where each cycle of the permutation is given a nonnegative constant multiplicative weight. A further contiguity result is proved involving the union of a weighted permutation pseudograph and a random regular graph of arbitrary degree. All corresponding results for simple graphs are obtained as corollaries.