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Graphs with many copies of a given subgraph

By Vladimir Nikiforov


Let c> 0, and H be a fixed graph of order r. Every graph on n vertices containing at least cnr copies of H contains a “blow-up ” of H with r − 1 vertex classes of size ⌊ cr2 ln n ⌋ and one vertex class of size greater than n1−cr−1. A similar result holds for induced copies of H. Main results Suppose that a graph G of order n contains cnr copies of a given subgraph H on r vertices. How large “blow-up ” of H must G contain? When H is an r-clique, this question was answered in [3]: G contains a complete r-partite graph with r − 1 parts of size ⌊ cr ln n ⌋ and one part larger than n1−cr−1. The aim of this note is to answer this question for any subgraph H. First we define precisely a “blow-up ” of a graph: given a graph H of order r and positive integers x1,..., xr, we write H(x1,..., xr) for the graph obtained by replacing each vertex u ∈ V (H) with a set Vu of size xu and each edge uv ∈ E(H) with a complete bipartite graph with vertex classes Vu and Vv

Year: 2010
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