Potential Bisections of Large Degree

A bisection of a graph G is a balanced bipartite spanning subgraph of G. Given a graphic sequence π, we show that π has a realization G containing a bisection H where degH(v) ≥ b(degG(v)− 1)/2c for all vertices v. This bound is very close to best possible. We use this result to provide evidence for a conjecture of Busch et. al. [1] that if π and π − k are graphic sequences, then π has a realization containing k edgedisjoint 1-factors. We show that if the minimum entry δ in π is at least n/2 + 2, then π has a realization containing ⌊ δ−2+ √ n(2δ−n−4) 4 ⌋ edge-disjoint 1-factors. We also give a construction showing the limits of our approach in proving this conjecture. 2010 AMS Math Subject Classification: 05C07