  For discrete weighted graphs there is sufficient literature about the Cheeger
cut and the Cheeger problem, but for metric graphs there are few results about
these problems. Our aim is to study the Cheeger cut and the Cheeger problem in
metric graphs. For that, we use the concept of total variation and perimeter in
metric graphs introduced in \cite{Mazon}, which takes into account the jumps at
the vertices of the functions of bounded variation. Moreover, we study the
eigenvalue problem for the minus $1$-Laplacian operator in metric graphs,
whereby we give a method to solve the optimal Cheeger cut problem.
