Counting maximal near perfect matchings in quasirandom and dense graphs

Abstract

A maximal $\varepsilon$-near perfect matching is a maximal matching which covers at least $(1-\varepsilon)|V(G)|$ vertices. In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs. We provide tight lower and upper bounds on the number of $\varepsilon$-near perfect matchings in generalized quasirandom graphs. Moreover, based on these results, we provide a deterministic polynomial time algorithm that for a given dense graph $G$ of order $n$ and a real number $\varepsilon>0$, returns either a conclusion that $G$ has no $\varepsilon$-near perfect matching, or a positive non-trivial number $\ell$ such that the number of maximal $\varepsilon$-near perfect matchings in $G$ is at least $n^{\ell n}$. Our algorithm uses algorithmic version of Szemer\'edi Regularity Lemma, and has $O(f(\varepsilon)n^{5/2})$ time complexity. Here $f(\cdot)$ is an explicit function depending only on $\varepsilon$.