The mass density of a spherical galaxy varies as $$\frac{K}{r}$$ over a large distance $$r$$ from its center. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as:

$$M(R) = \int_{0}^{R} \rho(r) \cdot 4\pi r^2 \, dr$$

$$M(R) = \int_{0}^{R} \frac{K}{r} \cdot 4\pi r^2 \, dr = 4\pi K \int_{0}^{R} r \, dr$$

$$M(R) = 4\pi K \left[ \frac{r^2}{2} \right]_{0}^{R} = 2\pi K R^2 \implies M(R) \propto R^2$$

$$\frac{m v^2}{R} = \frac{G M(R) m}{R^2}$$ $$\implies$$ $$v^2 = \frac{G M(R)}{R}$$

$$v^2 \propto \frac{R^2}{R} \implies v^2 \propto R \implies v \propto R^{1/2}$$

$$T = \frac{2\pi R}{v}$$ $$\implies$$ $$T^2 = \frac{4\pi^2 R^2}{v^2}$$

$$T^2 \propto \frac{R^2}{R} \implies T^2 \propto R$$