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A new lower bound for the inducibility of a graph. (English) Zbl 0612.05038
Let G be a p-vertex graph. For an arbitrary n-vertex graph H (n\(\geq p)\), let \({\mathcal I}(G,H)\) denote the number of induced subgraphs of H that are isomorphic to G. Let I(G,n) denote the maximum of \({\mathcal I}(G,H)/\left( \begin{matrix} n\\ p\end{matrix} \right)\) taken over all n-vertex graphs H. M. C. Pippenger and the reviewer [J. Comb. Theory, Ser. B 19, 189-203 (1975; Zbl 0332.05119)] have shown that the sequence I(G,n) is nonincreasing and have defined the inducibility of G to be the limit \(I(G)=\lim_{n\to \infty}I(G,n)\). They proved a lower bound for the inducibility given by I(G)\(\geq p!(p^ p-p)^{-1}.\)
In the paper under review the author gives a more general formula for the lower bound of I(G) which includes the previous bound as a special case and is an improvement upon previous bounds for an infinite family of graphs.

MSC:
05C35 Extremal problems in graph theory
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References:
[1] BERGE, C: Graphs and Hypergraphs. North Holland, Amsterdam-London, 1973, 396-397.
[2] HARARY F.: Graph Theory. Addison-Wesley, Reading, Mass., 1969. · Zbl 0182.57702
[3] PIPPENGER N., GOLUMBIC M. CH.: The Inducibility of a Graph. J. Comb. Theory (B)19, 1975, 189-203. · Zbl 0332.05119
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