Quasimonophobic graphs and degree spectral sequences in discrete cubical homology

TL;DR

This paper introduces a new degree filtration and spectral sequence in discrete cubical homology of graphs, linking it to injective homology and exploring quasimonophobic graphs.

Contribution

It defines quasimonophobicity, showing it causes spectral sequence vanishing and relates injective homology to CW complex homology, advancing graph homology theory.

Findings

Degree spectral sequence interpolates between homologies.

Quasimonophobicity implies spectral sequence vanishing.

Computed homology for Greene sphere graphs.

Abstract

We introduce the degree filtration on the discrete cubical chain complex of a graph, defined in terms of the maximal injective dimension of the facets of singular $n$-cubes, and study the degree spectral sequence which arises from this filtration. This spectral sequence interpolates between the discrete cubical homology of a graph $H_n(G)$ and the injective homology $H_n^{inj}(G)$, a variant of the discrete cubical homology based on injective singular cubes. Building on the work of Babson et al. we introduce the combinatorial condition of quasimonophobicity on graphs, and show quasimonophobicity implies both the vanishing of the degree spectral sequence in certain bidegrees, and implies $H_n^{inj}(G)$ is isomorphic to the homology of the CW complex obtained by ``filling in'' subcubes of the graph. These results are applied to compute $H_2(G_n^{sph})$ for the Greene sphere graphs $G^{sph}_n$.