Stability for large forbidden subgraphs

Abstract

In this note we strengthen the stability theorem of Erdös and Simonovits.Write Kr(s1, . . . ,sr) for the complete r-partite graph with classes of sizes s1, . . . ,s r and Tr(n) for the r-partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, e>0, every graph G of sufficiently large order n with e(G)>(1-1/ r-ε)n 2 / 2 satisfies one of the conditions: (a) G contains a K r+1([clnn], . . . , [clnn],[n1-√c]); (b) G differs from T r(n) in fewer than (ε1/3+c1/(3r+3))n 2 edges. Letting μ(G) be the spectral radius of G, we prove also a spectral stability theorem: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with μ(G)>(1-1/ r-ε)n satisfies one of the conditions: (a) G contains a K r+1([clnn], . . . ,[clnn],[n1-√c]); (b) G differs from T r(n) in fewer than (ε1/4+c1/(8r+8))n 2 edges. © 2009 Wiley Periodicals, Inc.

Publication Title

Journal of Graph Theory

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