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Monochromatic cycle partitions in random graphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  14 August 2020

Richard Lang  and
Allan Lo
Richard Lang*
Affiliation:
Institute for Computer Science, Heidelberg University, 69120Heidelberg, Germany
Allan Lo
Affiliation:
School of Mathematics, University of Birmingham, BirminghamB15 2TT, UK
*
*Corresponding author. Email: lang@informatik.uni-heidelberg.de
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Abstract

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Erdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = \Omega(n^{-1/(2r)})$ , then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 05C38: Paths and cycles
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 1 , January 2021 , pp. 136 - 152
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Footnotes

†

The research leading to these results was supported by EPSRC, grant EP/P002420/1.

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