Monotone cohomologies and oriented matchings

TL;DR

This paper extends cohomology theories for monotone graph properties to include twisted coefficients, introduces oriented matchings, and links their cohomology to multipath cohomology, providing new topological and combinatorial insights.

Contribution

It introduces oriented matchings on graphs, characterizes them via free-flow pseudoforests, and connects their cohomology to multipath cohomology, expanding the theoretical framework.

Findings

Characterization of oriented matchings as free-flow pseudoforests

Explicit determination of the homotopy type of associated complexes

Connection between cohomology of oriented matchings and multipath cohomology

Abstract

In this paper, we extend the definition of cohomology associated to monotone graph properties, to encompass twisted functor coefficients. We introduce oriented matchings on graphs, and focus on their (twisted) cohomology groups. We characterise oriented matchings in terms of induced free-flow pseudoforests, and explicitly determine the homotopy type of the associated simplicial complexes. Furthermore, we provide a connection between the cohomology of oriented matchings with certain functor coefficients, and the recently defined multipath cohomology. Finally, we define a further oriented homology for graphs and interpret it as a count of free-flow orientations.