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Karp–Sipser on Random Graphs with a Fixed Degree Sequence

Published online by Cambridge University Press:  20 June 2011

TOM BOHMAN  and
ALAN FRIEZE
TOM BOHMAN
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@math.cmu.edu, alan@random.math.cmu.edu)
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: tbohman@math.cmu.edu, alan@random.math.cmu.edu)
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Abstract

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +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,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 721 - 741
Copyright
Copyright © Cambridge University Press 2011

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