Let G be a graph, and let k ≥ 1 be an integer. Let U be a subset of V(G), and let F be a spanning subgraph of G such that deg_F(x) = k for all x ∈ V(G) - U. If deg_F(x) ≥ k for all x ∈ U, then F is called an upper semi-k-regular factor with defect set U, and if deg_F(x) ≤ k for all x ∈ U, then F is called a lower semi-k-regular factor with defect set U. We show that if k|V(G)| is even, |V(G)| ≥ k + 2, and for any subset U of cardinality k + 2 of V(G), G has an upper semi-k-regular factor with defect set U, then G has a k-factor. We also show that if k is even, |V(G)| ≥ 2k + 4, and for any subset U of cardinality k + 3 of V(G), G has an upper semi-k-regular factor with defect set U, then G has a k-factor. Further, we show that if k|V(G)| is even, |V(G)| ≥ k + 4, and for any subset U of cardinality 3 of V(G), G has a lower semi-k-regular factor with defect set U, then G has a k-factor.
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