In , Faudree et al. considered the proposition “Every (X, Y)-free graph of sufficiently large order has a 2-factor,” and they determined those pairs (X, Y) which make this proposition true. Their result says that one of them is (X, Y) = (K1,4, P4). In this paper, we investigate the existence of 2-factors in r-connected (K1, k, P4)-free graphs. We prove that if r ≥ 1 and k ≥ 2, and if G is an r-connected (K1, k, P4)-free graph with minimum degree at least k − 1, then G has a 2-factor with at most max(k − r, 1) components unless (k − 1)K2 + (k − 2)K1 ⊆ G ⊆ (k − 1)K2 + Kk−2. The bound on the minimum degree is best possible.
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