Edge-critical subgraphs of Schrijver graphs II: The general case

TL;DR

This paper provides a simple combinatorial description of a specific edge-critical subgraph of Schrijver graphs, extending previous results to all parameters and refining classical bounds from the 1970s.

Contribution

It generalizes the description of edge-critical subgraphs of Schrijver graphs for all k, building on and sharpening previous classical results.

Findings

Provides a combinatorial characterization of edge-critical subgraphs

Extends previous results to all values of k

Sharpens classical bounds from Lovász and Schrijver

Abstract

We give a simple combinatorial description of an $(n-2k+2)$-chromatic edge-critical subgraph of the Schrijver graph $\mathrm{SG}(n,k)$, itself an induced vertex-critical subgraph of the Kneser graph $\mathrm{KG}(n,k)$. This extends the main result of [J. Combin. Theory Ser. B 144 (2020) 191--196] to all values of $k$, and sharpens the classical results of Lov\'asz and Schrijver from the 1970s.