Every d-regular graph on n vertices with d ≤ n^(1/12-ε) has a spanning subgraph realizing any admissible degree-count vector q with |q_i - n/(d+1)| ≤ 1.
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Irregular subgraph in a regular graph
Every d-regular graph on n vertices with d ≤ n^(1/12-ε) has a spanning subgraph realizing any admissible degree-count vector q with |q_i - n/(d+1)| ≤ 1.