@MISC{Nikiforov06cyclesand, author = {V. Nikiforov and R. H. Schelp}, title = {Cycles and Stability (full version)}, year = {2006} }
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Abstract
We prove a number of Turán and Ramsey type stability results for cycles, in particular, the following one: Let n 4, 0 < 1=2 1=2n, and the edges of Kb(2 )nc be 2-colored so that no mononchromatic Cn exists. Then, for some q 2 ((1 )n 1; n) ; we may drop a vertex v so that in Kb(2 )nc v one of the colors induces Kq;b(2 )ncq1; while the other one induces Kq [Kb(2 )ncq1: We also derive the following Ramsey type result. If n is su ¢ ciently large and G is a graph of order 2n 1; with minimum degree (G) 2 106n; then for every 2-coloring of E (G) one of the colors contains cycles Ct for all t 2 [3; n]: 1