  A complexity-one space is a compact symplectic manifold $(M, \omega)$ endowed
with an effective Hamiltonian action of a torus $T$ of dimension
$\frac{1}{2}\dim(M)-1$. In this note we prove that for a certain class of
complexity-one spaces the Poincar\'e dual of the Chern class $c_{n-1}$ can be
represented by a collection of $\frac{n}{2}\chi(M)$ symplectic embedded
$2$-spheres, where $\chi(M)$ is the Euler characteristic of $M$ and
$\dim(M)=2n$. We call such a collection a toric one-skeleton. The
classification of complexity-one spaces is an important subject in symplectic
geometry. A nice subcategory of those spaces are the ones which are monotone.
The existence of a toric one-skeleton is a useful tool to understand
six-dimensional monotone complexity-one spaces. In particular, we will show
that the existence of a toric one-skeleton for such a space implies that the
second Betti number of $M$ is at most seven. This is a simple application of
results by Sabatini-Sepe and Lindsay-Panov.
