Contract grant sponsor: Nebraska EPSCoR First Award and NSF (to S.G.H.); Contract grant sponsor: NSF; Contract grant number: CCF-0916525 (to D.S.) and DMS-0914815 (to S.G.H. and D.S.); Contract grant sponsor: NSA; Contract grant number: H98230-10-1-0363 (to D.B.W.); Contract grant sponsor: NSF; Contract grant number: DMS 08-38434 (to M.Y.).
Extremal Graphs With a Given Number of Perfect Matchings
Article first published online: 16 JUL 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 73, Issue 4, pages 449–468, August 2013
How to Cite
Hartke, S. G., Stolee, D., West, D. B. and Yancey, M. (2013), Extremal Graphs With a Given Number of Perfect Matchings. J. Graph Theory, 73: 449–468. doi: 10.1002/jgt.21687
- Issue published online: 4 JUN 2013
- Article first published online: 16 JUL 2012
- Manuscript Revised: 28 MAY 2012
- Manuscript Received: 19 MAY 2011
- Nebraska EPSCoR First Award and National Science Foundation
- NSF. Grant Number: CCF-0916525 DMS-0914815
- National Security Agency. Grant Number: H98230-10-1-0363
- NSF. Grant Number: DMS 08-38434
- perfect matching;
- Lovász Cathedral Theorem;
- Hetyei's Theorem;
Let denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that for some constant when n is at least some constant . For , they also determined and . For fixed p, we show that the extremal graphs for all n are determined by those with vertices. As a corollary, a computer search determines and for . We also present lower bounds on proving that for (as conjectured by Dudek and Schmitt), and we conjecture an upper bound on . Our structural results are based on Lovász's Cathedral Theorem.