A Fan-Type Result for Regular Factors

Abstract

Let $G$ be a connected graph of order $n$ and let $k$ be a positive integer with $kn$ even and $n\ge 8k^2+12k+6$. We show that if $\delta (G)\ge k$ and $max\{d(u),d(v)\}\ge n/2$ 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.