A Fan-Type Result for Regular Factors

Let \(G\) be a connected graph of order \(n\) and let \(k\) be a positive integer with \(kn\) even and \(n \geq 8k^2 + 12k + 6\). We show that if \(\delta(G) \geq k\) and \(\max\{d(u), d(v)\} \geq n/2\) for each pair of vertices \(u,v\) at distance two, then \(G\) has a \(k\)-factor. Thereby a conjecture of Nishimura is answered in the affirmative.