@article{CHENG2022, 
author = {Shuting CHENG and Baoyindureng WU},
title = {Monochromatic Cycles and Trees in Edge-Colored Complete Graphs},
year = {2022},
journal = {Journal of Xinjiang University(Natural Science Edition in Chinese and English)},
volume = {39},
number = {1},
pages = {16-18,41},
keywords = {circumference, edge-colored complete graphs, monochromatic cycles, monochromatic trees},
url = {https://www.sciopen.com/article/10.13568/j.cnki.651094.651316.2021.12.08.0001},
doi = {10.13568/j.cnki.651094.651316.2021.12.08.0001},
abstract = {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  nr for r ≥ 2. Meanwhile, they also proved that f(2,n) ≥  ⌈2n3⌉ for n ≥ 6, and this bound is sharp. In 2011, Fujita disproved this conjecture for n=2r and also showed that every r-edge-colored complete graph Kn contains a monochromatic cycle of length at least  nr for 1 ≤ r ≤ n. In this paper, we disprove this conjecture for n=rt+1, where r ≥ 2 and  n−1r 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  nr. 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) =  ⌈n2⌉ if n ≡ 0,1 (mod 4) and moc(n, 3) =  ⌈n+12⌉ if n ≡ 2,3 (mod 4).}
}