Schrijver graphs and projective quadrangulations

TL;DR

This paper proves a conjecture linking Schrijver graphs to quadrangulations of projective spaces, extending topological graph theory concepts and demonstrating specific chromatic properties.

Contribution

It establishes that Schrijver graphs contain spanning subgraphs that are quadrangulations of certain projective spaces, confirming a previously conjectured relationship.

Findings

Schrijver graphs contain quadrangulations of projective spaces.

The conjecture by the authors has been proven.

The result links graph coloring with topological surface properties.

Abstract

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space $P^n$ is at least $(n+2)$-chromatic, unless it is bipartite. They conjectured that for any integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $SG(n,k)$ contains a spanning subgraph which is a quadrangulation of $P^{n-2k}$. The purpose of this paper is to prove the conjecture.