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

Abstract

We give a simple combinatorial description of an (n-2k+2)-chromatic edge-critical subgraph of the Schrijver graph SG(n,k), itself an induced vertex-critical subgraph of the Kneser graph KG(n,k). This extends the main result of Kaiser and Stehlík (2020) to all values of k, and sharpens the classical results of Lovász and Schrijver from the 1970s.