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Complexes of graph homomorphisms. (English) Zbl 1205.52009
For any two graphs \(G\) and \(H\) there is a polyhedral complex \({\operatorname{Hom}}(G,H)\), the Hom-complex of \(G\) and \(H\), whose vertices are the graph homomorphisms from \(G\) to \(H\). Especially interesting are the complexes \({\operatorname{Hom}}(G,K_n)\) where \(K_n\) is the complete graph on \(n\) nodes. This complex is non-empty if and only if \(G\) admits a proper coloring of its nodes with \(n\) colors.
In the paper under review the authors prove a number of combinatorial and topological results about Hom-complexes. In particular, they show that \({\operatorname{Hom}}(K_2,G)\) is homotopy equivalent to the neighborhood complex of \(G\). The degree of connectedness of the latter is known to form an obstruction to the colorability of \(G\) from L. Lovász’ solution of the Kneser conjecture [J. Comb. Theory, Ser. A 25, 319–324 (1978; Zbl 0418.05028)].
Moreover, it is proved that \({\operatorname{Hom}}(K_m,K_n)\) is homotopy equivalent to a wedge of \((n-m)\)-dimensional spheres, and a recurrence relation is given for their number. Other results involve the computation of homotopy types of Hom-complexes from finite forests to complete graphs. The techniques developed in this paper were applied in the authors’ proof [Ann. Math. (2) 165, No. 3, 965–1007 (2007; Zbl 1132.05019)] of a conjecture of Lovász.

MSC:
52B70 Polyhedral manifolds
05C15 Coloring of graphs and hypergraphs
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