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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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##### References:
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