This is related to Dirac's theorem.

For any finite, simple, undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimal degree of all vertices.

Are there positive integers $n,c\in\mathbb{N}$ with the following property?

Whenever $G=(V,E)$ is connected and $\delta(G)\geq n$, there is a matching $M\subseteq E$ such that $$|V\setminus \bigcup M|\leq c.$$

size of the smallest maximal matchinghas been studied in the literature, rather rarely. E.g. Vesna Andova, František Kardoš, Riste Škrekovski:Sandwiching Saturation Number of Fullerene Graphs. MATCH Commun. Math. Comput. Chem. 73 (2015) . (arxiv.org/abs/1405.2197) have some results for fullerene-graphs (they mention the general fact thatany graph $G$ has a maximal matching of size $\frac{n-\alpha(G)}{2}$). Since $\alpha\leq n-\delta$ in general, it follows that anygraph has a maximal matching of size $\geq\frac12\delta(G)$. $\endgroup$ – Peter Heinig Aug 7 '18 at 9:31