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The neighborhood complex of a random graph. (English) Zbl 1110.05090
Summary: For a graph \(G\), the neighborhood complex \(\mathcal {N}[G]\) is the simplicial complex having all subsets of vertices with a common neighbor as its faces. It is a well-known result of L. Lovász [J. Comb. Theory, Ser. A 25, 319–324 (1978; Zbl 0418.05028)] that if \(||\mathcal {N}[G]||\) is \(k\)-connected, then the chromatic number of \(G\) is at least \(k+3\). We prove that the connectivity of the neighborhood complex of a random graph is tightly concentrated, almost always between 1/2 and 2/3 of the expected clique number. We also show that the number of dimensions of nontrivial homology is almost always small, \(O(\log d)\), compared to the expected dimension \(d\) of the complex itself.

MSC:
05C80 Random graphs (graph-theoretic aspects)
05C15 Coloring of graphs and hypergraphs
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