  The chromatic alpha filtration is a generalization of the alpha filtration
that can encode spatial relationships among classes of labelled point cloud
data, and has applications in topological data analysis of multi-species data.
In this paper we introduce the chromatic Delaunay--\v{C}ech and chromatic
Delaunay--Rips filtrations, which are computationally favourable alternatives
to the chromatic alpha filtration. We use generalized discrete Morse theory to
show that the \v{C}ech, chromatic Delaunay--\v{C}ech, and chromatic alpha
filtrations are related by simplicial collapses. Our result generalizes a
result of Bauer and Edelsbrunner from the non-chromatic to the chromatic
setting. We also show that the chromatic Delaunay--Rips filtration is locally
stable to perturbations of the underlying point cloud. Our results provide
theoretical justification for the use of chromatic Delaunay--\v{C}ech and
chromatic Delaunay--Rips filtrations in applications, and we demonstrate their
computational advantage with numerical experiments.
