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A degree condition for the existence of k-factors

Let k be an integer such that k≥3, and let G be a connected graph of order n with n≥4k-3, kn even, and minimum degree at least K. We prove that it G satisfies max'deg(u), deg(v))≥n/2 for each pair of nonadjacent vertices u, v in G, then G has a k-factor