The neighborhood complex of a random graph

Details The neighborhood complex of a random graph The neighborhood complex of a random graph

2005-12-04

arxiv.org/abs/math/0512077v4

J. Combin. Theory Ser. A 114 (2007), no. 2, 380--387

Mathematics

Combinatorics

9 pages; stated theorems more clearly and slightly generalized, and fixed one or two typos

Scientific paper

For a graph G, the neighborhood complex 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 Lovasz that if 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.

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