Cell structure of bipartite mediangle graphs

TL;DR

This paper proves that bipartite mediangle graphs can be structured as contractible cell complexes, specifically as tope graphs of finitary Complexes of Oriented Matroids, linking graph theory with oriented matroid theory.

Contribution

It establishes that bipartite mediangle graphs are tope graphs of finitary COMs and characterizes the oriented matroids involved as simplicial OMs.

Findings

Bipartite mediangle graphs are tope graphs of finitary COMs.

The oriented matroids in these COMs are exactly the simplicial OMs.

Provides a positive answer to Genevois's question about the cell complex structure.

Abstract

Genevois introduced and investigated mediangle graphs as a common generalization of median graphs (1-sekeleta of CAT(0) cube complexes) and Coxeter graphs (Cayley graphs of Coxeter systems) and studied groups acting on them. He asked if mediangle graphs can be endowed with the structure of a contractible cell complex. We answer this in the affirmative by proving that bipartite mediangle graphs are tope graphs of finitary Complexes of Oriented Matroids (COMs). We also show that the oriented matroids (OMs) constituting the cells of COMs arising from bipartite mediangle graphs are exactly the simplicial OMs.