The universality of Hom complexes of graphs

Abstract

It is shown that given a connected graph T with at least one edge and an arbitrary finite simplicial complex X, there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma. Along the way several results regarding Hom complexes, exponentials of graphs, and subdivisions are established that may be of independent interest.

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Correspondence to Anton Dochtermann.

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Dochtermann, A. The universality of Hom complexes of graphs. Combinatorica 29, 433–448 (2009). https://doi.org/10.1007/s00493-009-2376-7

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Mathematics Subject Classification (2000)

  • 05C15
  • 55P15
  • 57M15