Long rainbow path in properly edge-colored complete graphs

Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let the color degree of a vertex $v$ be the number of different colors that are used on the edges incident to $v$, and denote it to be $d^c(v)$. It was shown that if $d^c(v)\ge k$ for every vertex $v$ of $G$, then $G$ has a rainbow path of length at least $\min \{\lceil \frac{2k+1}{3}\rceil ,k-1\}$. In the present paper, we consider the properly edge-colored complete graph $K_n$ only and improve the lower bound of the length of the longest rainbow path by showing that if $n\ge 20$, there must have a rainbow path of length no less than $\displaystyle \frac{3}{4}n-\frac{1}{4}\sqrt{\frac{n}{2}-\frac{39}{11}}-\frac{11}{16}$.