Some remarks on long monochromatic cycles in edge-colored complete graphs

In [On the circumference of a graph and its complement, Discrete Math. 309 (2009), 5891–5893], Faudree et al. conjectured that when r ≥ 3 , every r -edge-colored complete graph K n contains a monochromatic cycle of length at least n / ( r − 1 ) . We disprove this conjecture for small n and give a short proof of the following weaker but more generalized form: for r ≥ 1 , every r -edge-colored complete graph K n contains a monochromatic cycle of length at least n / r .