I am looking for a simple degree conditon that ensures the existence of a k-factor in a graph. The k is supposed to be relatively high and I don't mind the condition being a bit strict. Ideally, something of the form $\delta(G) \geq f(k)$. Any suggestions? 10x!

To clarify a bit what I'm after: there is a theorem by Nishimura that ensures a k-factor for k not larger than n/4 or so. But I want a k-factor with k approaching n.

http://onlinelibrary.wiley.com/doi/10.1002/jgt.3190160205/abstract