Israel Journal of Mathematics

, Volume 187, Issue 1, pp 371–417 | Cite as

Topology of Hom complexes and test graphs for bounding chromatic number

  • Anton Dochtermann
  • Carsten Schultz


The Hom complex of homomorphisms between two graphs was originally introduced to provide topological lower bounds on the chromatic number. In this paper we introduce new methods for understanding the topology of Hom complexes, mostly in the context of Γ-actions on graphs and posets (for some group Γ). We view the Hom(T, ⊙) and Hom(⊙, G) complexes as functors from graphs to posets, and introduce a functor ()1 from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of Hom complexes in terms of spaces of equivariant poset maps and Γ-twisted products of spaces. When P:= F(X) is the face poset of a simplicial complex X, this provides a useful way to control the topology of Hom complexes. These constructions generalize those of the second author from [17] as well as the calculation of the homotopy groups of Hom complexes from [8].

Our foremost application of these results is the construction of new families of test graphs with arbitrarily large chromatic number—graphs T with the property that the connectivity of Hom(T,G) provides the best possible lower bound on the chromatic number of G. In particular we focus on two infinite families, which we view as higher dimensional analogues of odd cycles. The family of spherical graphs have connections to the notion of homomorphism duality, whereas the family of twisted toroidal graphs lead us to establish a weakened version of a conjecture (due to Lovász) relating topological lower bounds on the chromatic number to maximum degree. Other structural results allow us to show that any finite simplicial complex X with a free action by the symmetric group S n can be approximated up to S n -homotopy equivalence as Hom(K n ,G) for some graph G; this is a generalization of the results of Csorba from [5] for the case of n = 2. We conclude the paper with some discussion regarding the underlying categorical notions involved in our study.


Simplicial Complex Chromatic Number Test Graph Barycentric Subdivision Graph Homomorphism 
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  1. [1]
    E. Babson and D. N. Kozlov, Group actions on posets, Journal of Algebra 285 (2005), 439–450.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    E. Babson and D. N. Kozlov, Complexes of graph homomorphisms, Israel Journal of Mathematics 152 (2006), 285–312.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    E. Babson and D. N. Kozlov, Proof of the Lovász conjecture, Annals of Mathematics 165 (2007), 965–1007.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    G. R. Brightwell and P. Winkler, Graph homomorphisms and long range action, in Graphs, Morphisms and Statistical Physics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 63, American Mathematical Society, Providence, RI, 2004, pp. 29–47.Google Scholar
  5. [5]
    P. Csorba, Homotopy types of box complexes, Combinatorica 27 (2007), 669–682.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    P. Csorba, Fold and Mycielskian on homomorphism complexes, Contributions to Discrete Mathematics 3 (2008), 1–8.MathSciNetzbMATHGoogle Scholar
  7. [7]
    A. Dochtermann, Hom complexes and homotopy theory in the category of graphs, European Journal of Combinatorics 30 (2009), 490–509.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    A. Dochtermann, Homotopy groups of Hom complexes of graphs, Journal of Combinatorial Theory. Series A 116 (2009), 180–194.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    A. Dochtermann, The universality of Hom complexes, Combinatorica 29 (2009), 433–448.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. Gyárfás, T. Jensen and M. Stiebitz, On graphs with strongly independent color-classes, Journal of Graph Theory 46 (2004), 1–14.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    S. Hoory and N. Linial, A counterexample to a conjecture of Björner and Lovász on the χ-coloring complex, Journal of Combinatorial Theory. Series B 95 (2005), 346–349.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    G. M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series, Vol. 64, Cambridge University Press, Cambridge, 1982.Google Scholar
  13. [13]
    D. N. Kozlov, Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes, in Geometric Combinatorics, IAS/Park City Mathematics Series, Vol. 13, American Mathematical Society, Providence, RI, 2007, pp. 249–315.Google Scholar
  14. [14]
    L. Lovász, Kneser’s conjecture, chromatic number, and homotopy, Journal of Combinatorial Theory. Series A. 25 (1978), 319–324.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    J. Matoušek, Using the Borsuk-Ulam Theorem, Universitext, Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler.zbMATHGoogle Scholar
  16. [16]
    J. Nešetřil and C. Tardif, Duality theorems for finite structures (characterising gaps and good characterisations), Journal of Combinatorial Theory. Series B 80 (2000), 80–97.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    C. Schultz, Graph colorings, spaces of edges and spaces of circuits, Advances in Mathematics 221 (2009), 1733–1756.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    C. Schultz, A short proof of w 1n (Hom(C r+1,K n+2)) = 0 for all n and a graph colouring theorem by Babson and Kozlov, Israel Journal of Mathematics 170 (2009), 125–134.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    G. Simonyi and G. Tardos, Local chromatic number, Ky Fan’s theorem and circular colorings, Combinatorica 26 (2006), 587–626.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    C. Tardif, Fractional chromatic numbers of cones over graphs, Journal of Graph Theory 38 (2001), 87–94.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    R. T. Živaljević, WI-posets, graph complexes and2 -equivalences, Journal of Combinatorial Theory. Series A 111 (2005), 204–223.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    R. T. Živaljević, Combinatorial groupoids, cubical complexes, and the Lovász conjecture, Discrete & Computational Geometry 41 (2009), 135–161.MathSciNetzbMATHCrossRefGoogle Scholar

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© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.Technische Universität Berlin, MA 6-2BerlinGermany

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