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On the multi-colored Ramsey numbers of cycles

Łuczak, Tomasz, Simonovits, Miklós and Skokan, Jozef (2011) On the multi-colored Ramsey numbers of cycles. Journal of Graph Theory, 69 (2). pp. 169-175. ISSN 0364-9024

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Abstract

For a graph L and an integer k≥2, Rk(L) denotes the smallest integer N for which for any edge-coloring of the complete graph KN by k colors there exists a color i for which the corresponding color class contains L as a subgraph. Bondy and Erdo″s conjectured that, for an odd cycle Cn on n vertices, Rk(Cn)=2 k-1(n-1)+1 for n>3. They proved the case when k = 2 and also provided an upper bound Rk(Cn)≤(k+ 2)!n. Recently, this conjecture has been verified for k = 3 if n is large. In this note, we prove that for every integer k≥4, Rk(Cn≤ k2kn+o «n» as n → ∞ When n is even, Sun Yongqi, Yang Yuansheng, Xu Feng, and Li Bingxi gave a construction, showing that Rk(C n≥(k-1)n-2k+ 4. Here we prove that if n is even, then R k(Cn≤kn+o(n) as n→∞.

Item Type: Article
Official URL: http://onlinelibrary.wiley.com/journal/10.1002/(IS...
Additional Information: © 2011 Wiley-Blackwell
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Departments > Mathematics
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Date Deposited: 12 Jan 2012 14:33
URL: http://eprints.lse.ac.uk/41376/

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