Graphic Sequences Have Realizations Containing Bisections of Large Degree

A bisection of a graph is a balanced bipartite spanning subgraph. Bollobás and Scott conjectured that every graphG has a bisectionH such that degH(v) ≥ bdegG(v)/2c for all vertices v. We prove a degree sequence version of this conjecture: 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 Brualdi [2] and Busch et al. [3], that if π and π− k are graphic sequences, then π has a realization containing k edge-disjoint 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