$0–1$ incidence matrix is totally unimodular then $G$ is bipartite.

Question

I know the result incidence matrix of a bipartite graph is totally unimodular.

But I am stuck with a converse statement:

Let $Q$ be the $0–1$ incidence matrix of the graph $G$, if $Q$ is totally unimodular then $G$ is bipartite.

Require some hints to do the problem.

Answer 1

Hint: Show that the incidence matrix of an odd cycle is not totally unimodular.