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Complexes of graph homomorphisms

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Abstract

Hom(G, H) is a polyhedral complex defined for any two undirected graphsG andH. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes.

We prove that Hom(K m, Kn) is homotopy equivalent to a wedge of (n−m)-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graphG, and integersm≥2 andk≥−1, we have ϖ k1 (Hom(K m, G))≠0, thenχ(G)≥k+m; here ℤ2-action is induced by the swapping of two vertices inK m, and ϖ1 is the first Stiefel-Whitney class corresponding to this action.

Furthermore, we prove that a fold in the first argument of Hom(G, H) induces a homotopy equivalence. It then follows that Hom(F, K n) is homotopy equivalent to a direct product of (n−2)-dimensional spheres, while Hom(F, K n) is homotopy equivalent to a wedge of spheres, whereF is an arbitrary forest andF is its complement.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    Eric Babson

  2. Department of Computer Science, Eidgenössische Technische Hochschule IFW B27.2, Haldeneggsteig 4, CH-8092, Zürich, Switzerland

    Dmitry N. Kozlov

Authors
  1. Eric Babson
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  2. Dmitry N. Kozlov
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Corresponding author

Correspondence to Eric Babson.

Additional information

The second author acknowledges support by the University of Washington, Seattle, the Swiss National Science Foundation Grant PP002-102738/1, the University of Bern, and the Royal Institute of Technology, Stockholm.

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Babson, E., Kozlov, D.N. Complexes of graph homomorphisms. Isr. J. Math. 152, 285–312 (2006). https://doi.org/10.1007/BF02771988

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  • DOI: https://doi.org/10.1007/BF02771988

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