Further Remarks on Long Monochromatic Cycles in Edge-Colored Complete Graphs

Abstract

In [Discrete Math., 311 (2011), 688-689], Fujita defined $f(r,n)$ to be the maximum integer $k$ such that every $r$-edge-coloring of $K_n$ contains a monochromatic cycle of length at least $k$. In this paper, we investigate the values of $f(r,n)$ when $n$ is linear in $r$. We determine the value of $f(r,2r+2)$ for all $r\ge 1$ and show that $f(r,sr+c)=s+1$ if $r$ is sufficiently large compared with positive integers $s$ and $c$.