SIMPLE Σ_r-HOMOTOPY TYPES OF HOM COMPLEXES AND BOX COMPLEXES ASSIGNED TO r-GRAPHS

抄録

For a pair of graphs, Lovász introduced a polytopal complex called the Hom complex in order to estimate topological lower bounds for chromatic numbers of graphs. The definition is generalized to hypergraphs. Given an r-graph H , we compare the Hom complex of the complete r-partite r-graph and H with the box complex of H invented by Alon, Frankl and Lovász.We verify that both complexes which are equipped with right actions of the symmetric group on r letters Σ_r, are of the same simple Σ_r-homotopy type.