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On the multi-colored Ramsey numbers of cycles
Article first published online: 16 JAN 2011
DOI: 10.1002/jgt.20572
© 2011 Wiley Periodicals, Inc.
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How to Cite
Łuczak, T., Simonovits, M. and Skokan, J. (2012), On the multi-colored Ramsey numbers of cycles. J. Graph Theory, 69: 169–175. doi: 10.1002/jgt.20572
Publication History
- Issue published online: 15 DEC 2011
- Article first published online: 16 JAN 2011
- Manuscript Revised: 24 AUG 2010
- Manuscript Received: 22 APR 2010
Funded by
- Foundation for Polish Science )to T. Ł.( Hungarian National Science Foundation. Grant Numbers: OTKA T 026069, T 038210, T 0234702, T 69062 )to M. S.(
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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 Erdos conjectured that, for an odd cycle Cn on n vertices,
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,
When n is even, Sun Yongqi, Yang Yuansheng, Xu Feng, and Li Bingxi gave a construction, showing that Rk(Cn)≥(k−1)n−2k+ 4. Here we prove that if n is even, then
© 2011 Wiley Periodicals, Inc. J Graph Theory 69: 169-175, 2012