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

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.