Random Lifts Of Graphs: Perfect Matchings

We study random lifts of a graph G as defined in [1]. We prove a 0-1 law which states that for every graph G either almost every lift of G has a perfect matching, or almost none of its lifts has a perfect matching. We provide a precise description of this dichotomy. Roughly speaking, the a.s. existence of a perfect matching in the lift depends on the existence of a fractional perfect matching in G. The precise statement appears in Theorem 1.

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Correspondence to Nathan Linial*.

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* Supported in part by BSF and by the Israeli academy of sciences.

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Linial*, N., Rozenman, E. Random Lifts Of Graphs: Perfect Matchings. Combinatorica 25, 407–424 (2005). https://doi.org/10.1007/s00493-005-0024-8

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Mathematics Subject Classification (2000):

  • 05C80
  • 05C70