On the Number of Perfect Matchings and Hamilton Cycles in $\epsilon$-Regular Non-bipartite Graphs

Abstract

A graph $G=(V,E)$ on $n$ vertices is super $\epsilon$-regular if (i) all vertices have degree in the range $[(d-\epsilon)n,(d+\epsilon)n]$, $dn$ being the average degree, and (ii) for every pair of disjoint sets $S,T\subseteq V,\,|S|,|T|\geq \epsilon n$, $e(S,T)$ is in the range $[(d-\epsilon)|S||T|,(d+\epsilon)|S||T|]$. We show that the number of perfect matchings lies in the range $$[((d-2\epsilon)^\nu {{n!}\over {\nu !2^\nu}},(d+2\epsilon)^\nu {{n!}\over {\nu !2^\nu}}],$$ where $\nu ={{n}\over {2}}$, and the number of Hamilton cycles lies in the range $[(d-2\epsilon)^nn!,(d+2\epsilon)^nn!]$.