A Note on $\{k,n-k\}$-Factors of Regular Graphs
  Let $r$ be an odd integer, and $k$ an even integer. In this note, we present $r$-regular graphs which have no $\{k,r-k\}$-factors for all $1\le k\le {r\over2}-1$. This gives a negative answer to a problem posed by Akbari and Kano recently. 
Hongliang Lu and David G. L. Wang
math.CO