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Monochromatic cycles in 2-coloured graphs

Benevides, F. S., Łuczak, T., Scott, A., Skokan, Jozef and White, M. (2012) Monochromatic cycles in 2-coloured graphs. Combinatorics, Probability and Computing, 21 (1-2). pp. 57-87. ISSN 0963-5483

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Abstract

Li, Nikiforov and Schelp [13] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length ℓ, for all ℓ ∈ [4, ⌈n/2⌉]. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with δ(G)=3n/4 that do not contain all such cycles. Finally, we show that, for all δ>0 and n>n 0(δ), if G is a 2-edge coloured graph of order n with δ(G) ≥ 3n/4, then one colour class either contains a monochromatic cycle of length at least (2/3+δ/2)n, or contains monochromatic cycles of all lengths ℓ ∈ [3, (2/3-δ)n].

Item Type: Article
Official URL: http://journals.cambridge.org/action/displayJourna...
Additional Information: © 2012 Cambridge University Press
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: 19 Apr 2012 15:47
URL: http://eprints.lse.ac.uk/43289/

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