• degree sequence;
  • bisection;
  • k-factor;
  • 1-factor

A bisection of a graph is a balanced bipartite spanning sub-graph. Bollobás and Scott conjectured that every graph G has a bisection H such that degH(v) ≥ ⌊degG(v)/2⌋ 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) ≥ ⌊(degG(v) − 1)/2⌋ for all vertices v. This bound is very close to best possible. We use this result to provide evidence for a conjecture of Brualdi (Colloq. Int. CNRS, vol. 260, CNRS, Paris) and Busch et al. (2011), 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 inline image edge-disjoint 1-factors. We also give a construction showing the limits of our approach in proving this conjecture. © 2011 Wiley Periodicals, Inc. J Graph Theory