Monochromatic Cycles in 2-Coloured Graphs

Li, Nikiforov and Schelp [12] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length `, for all ` ∈ [4, dn/2e]. 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 a monochromatic cycle of all lengths ` ∈ [3, (2/3− δ)n].