Karp-Sipser on Random Graphs with a Fixed Degree Sequence

Let ∆ ≥ 3 be an integer. Given a fixed z ∈ R+ such that z∆ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1, . . . ,∆. 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, . . . ,∆z∆ 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− ) edges in Gz where = (∆, z) is a constant.