Generating p-extremal graphs

Let f(n, p) be the maximum number of edges in a graph on n vertices with p perfect matchings. Dudek and Schmitt proved that there exist constants np and cp so that for even n ≥ np, f(n, p) = n 2 4 + cp; we call a graph p-extremal if it has p perfect matchings and n2 4 + cp edges. In this paper, we develop structural theorems in matching theory to study p-extremal graphs and use them in a new computational method. This method extends the known sizes of p-extremal graphs from p ≤ 10 to p ≤ 27 while providing a complete characterization. This information provides further evidence towards a conjectured upper bound for all values of p and shows the sequence cp is not monotonic.