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Abstract:A bisection of a graph is a balanced bipartite spanning subgraph. Bollobás and Scott conjectured that every graph G has a bisection H such that deg H (v) ≥ deg G (v)/2 for all vertices v. We prove a degree sequence version of this conjecture: given a graphic sequence , we show that has a realization G containing a bisection H where deg H (v) ≥ (deg G (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 and −k are graphic sequences, then has a realization containing k edge-disjoint 1-factors. We show that if the minimum entry in is at least n /2+2, then has a realization containing ( −2+ √ n(2 −n−4))/4 edge-disjoint 1-factors. We also give a construction showing the limits of our approach in proving this conjecture.᭧

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