A spectral Erdo″s-Stone-Bolloba′s theorem


Let r ≥3 and (c/rr)r log n ≥1. If G is a graph of order n and its largest eigenvalue μ(G) satisfies μ(G) ≥ (1-1/(r-1 )+c)n, then G contains a complete r-partite subgraph with r - 1 parts of size ⌊(c/rr)r log n⌋ and one part of size greater than n1-cr-1. This result implies the Erdo″s-Stone-Bolloba′s theorem, the essential quantitative form of the Erdo″sStone theorem. Another easy consequence is that if F 1, F2,... are r-chromatic graphs satisfying v(F n) = o(log n), then lim n→∞ 1/n max{μ(G) : v(G) = n and Fn ⊈G} = 1-1/r-1. © 2009 Cambridge University Press.

Publication Title

Combinatorics Probability and Computing