Edge-critical subgraphs of Schrijver graphs

TL;DR

This paper constructs a new class of edge-critical subgraphs of Schrijver graphs for the case k=2, providing explicit combinatorial definitions and advancing understanding of graph criticality beyond vertex-criticality.

Contribution

It introduces an explicit combinatorial construction of edge-critical subgraphs of Schrijver graphs for k=2, filling a gap in the understanding of edge-criticality in these graphs.

Findings

Constructed edge-critical subgraphs for k=2 and n≥4.

Provided explicit combinatorial definitions for these subgraphs.

Extended the theory of graph criticality in Schrijver graphs.

Abstract

For $k\geq 1$ and $n\geq 2k$, the Kneser graph $KG(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lov\'{a}sz that the chromatic number of $KG(n,k)$ is $n-2k+2$. Schrijver constructed a vertex-critical subgraph $SG(n,k)$ of $KG(n,k)$ with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges, however, no analogous construction is known except in trivial cases. We provide such a construction for $k=2$ and arbitrary $n\geq 4$ by means of a nice explicit combinatorial definition.