  We introduce a filtration on the simplicial homology of a finite simplicial
complex X using bi-colourings of its vertices. This yields two dual homology
theories closely related to discrete Morse matchings on X. We give an explicit
expression for the associated graded object of these homologies when X is the
matching complex of the Tait graph of a plane graph $G$, in terms of subgraphs
determined by certain matchings on the dual of G. We then use one of these
homologies, in the case where X is a graph, to define a conjecturally optimal
dissimilarity pseudometric for graphs; we prove various results for this
dissimilarity and provide several computations. We further show that, by
organising the horizontal homologies of a simplicial complex in the poset of
its colourings, we obtain a triply graded homology theory which we call
\"uberhomology. This latter homology is not a homotopy invariant, but
nonetheless encodes both combinatorial and topological information on X. For
example, we prove that if X is a subdivision, the \"uberhomology vanishes in
its lowest degree, while for an homology manifold it coincides with the
fundamental class in its top degree. We compute the \"uberhomology on several
classes of examples and infinite families, and prove some of its properties;
namely that, in its extremal degrees, it is well-behaved under coning and
taking suspension. We then focus on the case where X is a simple graph, and
prove a detection result. Finally, we define some singly-graded homologies for
graphs obtained by specialising the \"uberhomology in certain bi-degrees,
provide some computations and use computer aided calculations to make some
conjectures.
