Monochromatic cycles and the monochromatic circumference in 2-coloured graphs

Details Monochromatic cycles and the monochromatic circumference in 2-coloured graphs Monochromatic cycles and the monochromatic circumference in 2-coloured graphs

2011-07-26

arxiv.org/abs/1107.5177v1

Mathematics

Combinatorics

35 pages, 2 figures. Submitted to CPC

Scientific paper

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with minimal degree equal to 3n/4 that do not contain all such cycles. Finally we show that, for all positive constants d and sufficiently large n, a 2-edge coloured graph G of order n with minimal degree at least 3n/4 either contains a monochromatic cycle of length at least (2/3+d/2)n, or, in one of the two colours, contains a cycle of all lengths between three and (2/3-d)n.

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