  Let $G$ be a graph whose edges are labeled by ideals of a commutative ring
$R$ with identity. Such a graph is called an edge-labeled graph over $R$. A
generalized spline is a vertex labeling so that the difference between the
labels of any two adjacent vertices lies in the ideal corresponding to the
edge. These generalized splines form a module over $R$. In this paper, we
consider complete graphs whose edges are labeled with proper ideals of
$\mathbb{Z} / m\mathbb{Z}$. We compute minimum generating sets of constant
flow-up classes for spline modules on edge-labeled complete graphs over
$\mathbb{Z} / m\mathbb{Z}$ and their rank under some restrictions.
