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Testing subgraphs in large graphs. (English) Zbl 1027.68095
Summary: Let \(H\) be a fixed graph with \(h\) vertices, let \(G\) be a graph on \(n\) vertices, and suppose that at least \(\varepsilon n^2\) edges have to be deleted from it to make it \(H\)-free. It is known that in this case \(G\) contains at least \(f(\varepsilon ,H)n^h\) copies of \(H\). We show that the largest possible function \(f(\varepsilon, H)\) is polynomial in \(\varepsilon\) if and only if \(H\) is bipartite. This implies that there is a one-sided error property tester for checking \(H\)-freeness, whose query complexity is polynomial in \(1/\varepsilon\), if and only if \(H\) is bipartite.

MSC:
68R10 Graph theory (including graph drawing) in computer science
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