Karp–Sipser on Random Graphs with a Fixed Degree Sequence

Let Δ â‰¥ 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,.i¾ .i¾ .,Δ. We study the performance of the Karp-Sipser algorithm when applied to Gz. If there is an index I´ > 1 such that z1 =.i¾ .i¾ . = zI´-1 = 0 and I´zI´,.i¾ .i¾ .,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-e) edges in Gz, where e = e (Δ, z) is a constant.

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