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- W4241969375 abstract "The classical Stability Theorem of Erdös and Simonovits can be stated as follows. For a monotone graph property P, let t≥2 be such that t+1=min{χ(H):H∉∈P}. Then any graph G*∈P on n vertices, which was obtained by removing at most edges from the complete graph G=Kn, has edit distance o(n2) to Tn(t), the Turán graph on n vertices with t parts. In this paper we extend the above notion of stability to hereditary graph properties. It turns out that to do so the complete graph Kn has to be replaced by a random graph. For a hereditary graph property P, consider modifying the edges of a random graph G=G(n,½) to obtain a graph G* that satisfies P in (essentially) the most economical way. We obtain necessary and sufficient conditions on P, which guarantee that G* has a unique structure. In such cases, for a pair of integers (r, s), which depends on P, G*, has distance o(n2) to a graph Tn(r, s, ½) almost surely. Here Tn(r, s, ½) denotes a graph, which consists of almost equal-sized r+s parts, r of them induce an independent set, s induce a clique and all the bipartite graphs between parts are quasi-random (with edge density ½. In addition, several strengthened versions of this result are shown. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 65–83, 2009" @default.
- W4241969375 created "2022-05-12" @default.
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- W4241969375 date "2009-09-01" @default.
- W4241969375 modified "2023-09-28" @default.
- W4241969375 title "Stability-type results for hereditary properties" @default.
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- W4241969375 doi "https://doi.org/10.1002/jgt.20388" @default.
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