Stability-type results for hereditary properties

Noga Alon, Uri Stav
2009 Journal of Graph Theory  
The classical Stability Theorem of Erdős 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 ( 1 t + o(1)) n 2 edges from the complete graph G = K n , has edit distance o(n 2 ) to T n (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
more » ... complete graph K n 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, 1/2) 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(n 2 ) to a graph T n (r, s, 1 2 ) almost surely. Here T n (r, s, 1 2 ) 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 1 2 ). In addition, several strengthened versions of this result are shown.
doi:10.1002/jgt.20388 fatcat:3odqiv2wnrbzzos6cg6aywn2fy