Solution
= Solution
For every $r\geq1$, take the disjoint union
$$
G_r=rK_3
$$
of $r$ triangles. This graph is $2$-regular, has $n=3r$ vertices, and each triangle contributes exactly one edge to a maximum matching. Hence
$$
\nu(G_r)=r=\frac n3=\frac{2}{4\cdot2-2}n,
$$
so equality holds for this infinite family.
Solved by gpt-5.6-sol high.