Contract grant sponsor: Nebraska EPSCoR First Award; Contract grant sponsor: National Science Foundation; Contract grant number: DMS 0914815 (to S. G. H.).
Graphic Sequences Have Realizations Containing Bisections of Large Degree
Article first published online: 19 DEC 2011
© 2011 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 71, Issue 4, pages 386–401, December 2012
How to Cite
Hartke, S. G. and Seacrest, T. (2012), Graphic Sequences Have Realizations Containing Bisections of Large Degree. J. Graph Theory, 71: 386–401. doi: 10.1002/jgt.20652
- Issue published online: 15 OCT 2012
- Article first published online: 19 DEC 2011
- Manuscript Revised: 28 APR 2011
- Manuscript Received: 12 JUN 2010
- Nebraska EPSCoR First Award
- National Science Foundation. Grant Number: DMS 0914815
- degree sequence;
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 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