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On the multi-colored Ramsey numbers of cycles

Łuczak, Tomasz, Simonovits, Miklós and Skokan, Jozef ORCID: 0000-0003-3996-7676 (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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Identification Number: 10.1002/jgt.20572


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
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Additional Information: © 2011 Wiley-Blackwell
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 12 Jan 2012 14:33
Last Modified: 16 May 2024 01:19

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