# Uniform generation of spanning regular subgraphs of a dense graph

Let H_n be a graph on n vertices and let H_n denote the complement of H_n. Suppose that Δ = Δ(n) is the maximum degree of H_n. We analyse three algorithms for sampling d-regular subgraphs (d-factors) of H_n. This is equivalent to uniformly sampling d-regular graphs which avoid a set E(H_n) of forbidden edges. Here d=d(n) is a positive integer which may depend on n. Two of these algorithms produce a uniformly random d-factor of H_n in expected runtime which is linear in n and low-degree polynomial in d and Δ. The first algorithm applies when (d+Δ)dΔ = o(n). This improves on an earlier algorithm by the first author, which required constant d and at most a linear number of edges in H_n. The second algorithm applies when H_n is regular and d^2+Δ^2 = o(n), adapting an approach developed by the first author together with Wormald. The third algorithm is a simplification of the second, and produces an approximately uniform d-factor of H_n in time O(dn). Here the output distribution differs from uniform by o(1) in total variation distance, provided that d^2+Δ^2 = o(n).

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