• perfect matching;
  • Lovász Cathedral Theorem;
  • matching-extendable;
  • Hetyei's Theorem;
  • chamber


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