Let $G$ be a simple graph with order $n$, number of edges $e$, and minimum degree $\delta<\frac{n}{2}$. If we have $e>\binom{\delta}{ 2}+\binom{n-2\delta-1}{2}+\delta(n-\delta)$. Then, will $G$ have a perfect matching?
I think yes, and I think using the characterization of Tutte's 1-factor theorem in this manner may be of some help:
A graph $G$ with order $n$ is a maximal non-matchable graph if and only if it has one of the following structures: (a)$n$ is odd and $G$ is complete, or, (b)$n$ is even and $G$ consists of vertex-disjoint complete subgraphs $S,G_1,...,G_k$ such that $k= |S|+2;G_1,...,G_k$ are odd, and every vertex of every $G_i$ is adjacent to every vertex of $S$.
Can the above lemma be used to prove the problem on hand? Any hints? Thanks beforehand.