Let f(r

n) be the maximum integer k such that every r-edge-colored complete graph Kn contains a monochromatic cycle of length at least k. In 2009

Faudree

Lesniak and Schiermeyer conjectured that every(r + 1)-edge-colored complete graph Kn contains a monochromatic cycle of length at least n/r for r ≥ 2. Meanwhile

they also proved that f(2

n) ≥ [2n/3] for n ≥ 6

and this bound is sharp. In 2011

Fujita disproved this conjecture for n = 2 r and also showed that every r-edge-colored complete graph Kn contains a monochromatic cycle of length at least n/r for 1 ≤ r ≤ n. In this paper

we disprove this conjecture for n = rt+1

where r ≥ 2 and n-1/r is a positive even integer. More precisely

there exists a(r + 1)-edge-colored complete graph Kn contains a monochromatic cycle of length less than n/r. For a k-edge coloring c of Kn

let moc(Kn

c) be the largest order of monochromatic tree of Kn under c. Let moc(n

k) = min{moc(Kn

c) : c is a k-edge coloring of Kn }. We show that for any positive integer n ≥ 3

moc(n

3) = [n/2] if n ≡ 0

1(mod 4) and moc(n

3) = [n+1/2] if n ≡ 2

3(mod 4).