  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 $d^c(v)\geq k$ 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 $n\geq 20$, there must have a rainbow path of length no less
than $\displaystyle
\frac{3}{4}n-\frac{1}{4}\sqrt{\frac{n}{2}-\frac{39}{11}}-\frac{11}{16}$.
