A regular graph G of degree r where G has n=3r vertices. Prove G has no Bridge

Question

A regular graph G of degree r where G has n=3r vertices. Prove G has no Bridge.

Does anyone have a hint?

Answer 1

Suppose $e$ is a bridge, so that $G-e$ has two components, $G_1$ and $G_2.$ Say $G_1$ has $n_1$ vertices and $G_2$ has $n_2$ vertices. So $G_1$ has $\frac{rn_1-1}2$ edges, and $G_2$ has $\frac{rn_2-1}2$ edges. So $rn_1$ and $rn_2$ are odd, and $n_1+n_2=3r.$ So $r,n_1,n_2$ are all odd, so $n_1+n_2$ is even, so $3r$ is even, so $r$ is even. . . oops!