Paths of homomorphisms from stable Kneser graphs

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≡3 (mod 4) and n≥2 we show that there is a component of the χ-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 χ(G)=χ(SG n,k ) such that the complex Hom(SG n,k ,G) is non-empty and connected. In particular, for k≡3 (mod 4) and n≥2 the graph SG n,k is not a test graph.

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Correspondence to Carsten Schultz.

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Schultz, C. Paths of homomorphisms from stable Kneser graphs. Combinatorica 33, 613–621 (2013). https://doi.org/10.1007/s00493-013-2749-9

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Mathematics Subject Classification (2010)

  • 05C15
  • 05C76
  • 05C40