Extremal Graphs With a Given Number of Perfect Matchings


  • 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.).


Let math formula 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 math formula for some constant math formula when n is at least some constant math formula. For math formula, they also determined math formula and math formula. For fixed p, we show that the extremal graphs for all n are determined by those with math formula vertices. As a corollary, a computer search determines math formula and math formula for math formula. We also present lower bounds on math formula proving that math formula for math formula (as conjectured by Dudek and Schmitt), and we conjecture an upper bound on math formula. Our structural results are based on Lovász's Cathedral Theorem.