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Monochromatic Cycles in 2-Coloured Graphs

Published online by Cambridge University Press:  19 March 2012

F. S. BENEVIDES
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Ceará, Brazil, 60455-760 (e-mail: fabricio@mat.ufc.br)
T. ŁUCZAK
Affiliation:
Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: tomasz@amu.edu.pl)
A. SCOTT*
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford, OX1 3LB, UK (e-mail: scott@maths.ox.ac.uk, white@maths.ox.ac.uk)
J. SKOKAN
Affiliation:
Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: jozef@member.ams.org)
M. WHITE
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford, OX1 3LB, UK (e-mail: scott@maths.ox.ac.uk, white@maths.ox.ac.uk)
*
§Corresponding author.

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>n0(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].

Type
Paper
Copyright
Copyright © Cambridge University Press 2012

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