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 Lovasz's Cathedral Theorem.

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Extremal Graphs With a Given Number of Perfect Matchings

and a University of Nebraska Presidential Fellowship.

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Combinatorics Using Computational Methods

Let be subsets of a set such that . For the configuration , let denote the number of SDR's of . Ostrand[Os] proved that if > 0, then . In this note we give a simple proof for this inequality.

Suk-Geun Hwang

Papers

A Note on System of Distinct Representatives