On factorable degree sequences

We call a degree sequence graphic (respectively, k-factorable, connected k-factorable) if there exists a graph (respectively, a graph with a k-factor, a graph with a connected k-factor) with the given degree sequence. In this paper we give a necessary and sufficient condition for a k-factorable sequence to be connected k-factorable when k ? 2. We also prove that every k-factorable sequence is (k − 2) factorable and 2-factorable, and also 1-factorable, when the sequence is of even length. Some conjectures are stated and it is also proved that, if {d_i} and {d_i − k} are graphic, then {d_i − r} is graphic for 0 ≤ r ≤ k provided rn is even.