Title

Cycles and stability

Abstract

We prove a number of Turán and Ramsey type stability results for cycles, in particular, the following one: Let n > 4, 0 < β ≤ 1 / 2 - 1 / 2 n, and the edges of K⌊ (2 - β) n ⌋ be 2-colored so that no monochromatic Cn exists. Then, for some q ∈ ((1 - β) n - 1, n), we may drop a vertex v so that in K⌊ (2 - β) n ⌋ - v one of the colors induces Kq, ⌊ (2 - β) n ⌋ - q - 1, while the other one induces Kq ∪ K⌊ (2 - β) n ⌋ - q - 1. We also derive the following Ramsey type result. If n is sufficiently large and G is a graph of order 2 n - 1, with minimum degree δ (G) ≥ (2 - 10-6) n, then for every 2-coloring of E (G) one of the colors contains cycles Ct for all t ∈ [3, n]. © 2007 Elsevier Inc. All rights reserved.

Publication Title

Journal of Combinatorial Theory. Series B

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