Karp-sipser on random graphs with a fixed degree sequence

  • Authors:

  • Tom Bohman;Alan Frieze

  • Affiliations:

  • Department of mathematical sciences, carnegie mellon university, pittsburgh, pa 15213, usa (e-mail: tbohman@math.cmu.edu, alan@random.math.cmu.edu);Department of mathematical sciences, carnegie mellon university, pittsburgh, pa 15213, usa (e-mail: tbohman@math.cmu.edu, alan@random.math.cmu.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2011

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Abstract

Let Δ â聣楼 3 be an integer. Given a fixed z â聢聢 +Δ such that zΔ 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,.ï戮 .ï戮 .,Δ. We study the performance of the Karp-Sipser algorithm when applied to Gz. If there is an index Î麓 1 such that z1 =.ï戮 .ï戮 . = zÎ麓-1 = 0 and Î麓zÎ麓,.ï戮 .ï戮 .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp-Sipser algorithm succeeds in finding a matching with n â聢楼 z â聢楼 1/2-o(n1-ε) edges in Gz, where ε = ε (Δ, z) is a constant.