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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
query complexity
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