Critical subgraphs of Schrijver graphs for the fractional chromatic number

TL;DR

This paper identifies specific subgraphs within Schrijver graphs that are vertex-critical for their fractional chromatic number, revealing their isomorphism to circular complete graphs and characterizing their critical edges.

Contribution

It introduces induced subgraphs of Schrijver graphs that are vertex-critical for fractional chromatic number and characterizes their critical edges.

Findings

Subgraphs are isomorphic to circular complete graphs.

These subgraphs are vertex-critical for fractional chromatic number.

Critical edges within these subgraphs are characterized.

Abstract

Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.