Paths of homomorphisms from stable Kneser graphs

TL;DR

This paper investigates the structure of homomorphism paths from stable Kneser graphs, revealing invariant components under automorphisms and implications for graph test properties in certain parameter regimes.

Contribution

It demonstrates the existence of invariant components in the chi-coloring graph of stable Kneser graphs for specific parameters, and shows these graphs are not test graphs.

Findings

Existence of an automorphism-invariant component in the coloring graph.

Construction of a graph G with equal chromatic number where Hom(SG_{n,k}, G) is non-empty and connected.

Stable Kneser graphs are not test graphs for certain parameters.

Abstract

We denote by SG_{n,k} the stable Kneser graph (Schrijver graph) of stable n-subsets of a set of cardinality 2n+k. For k congruent 3 (mod 4) and n\ge2 we show that there is a component of the \chi-colouring graph of SG_{n,k} which is invariant under the action of the automorphism group of SG_{n,k}. We derive that there is a graph G with \chi(G)=\chi(SG_{n,k}) such that the complex Hom(SG_{n,k}, G) is non-empty and connected. In particular, for k congruent 3 (mod 4) and n\ge2 the graph SG_{n,k} is not a test graph.