  Let $G$ be an edge colored graph. A {\it}{rainbow path} in $G$ is a path in
which all the edges are colored with distinct colors. Let $d^c(v)$ be the color
degree of a vertex $v$ in $G$, i.e. the number of distinct colors present on
the edges incident on the vertex $v$. Let $t$ be the maximum length of a
rainbow path in $G$. Chen and Li showed that if $d^c \geq k$, for every vertex
$v$ of $G$, then $t \geq \left \lceil \frac{3 k}{5}\right \rceil + 1$ (Long
heterochromatic paths in edge-colored graphs, The Electronic Journal of
Combinatorics 12 (2005), # R33, Pages:1-33.) Unfortunately, proof by Chen and
Li is very long and comes to about 23 pages in the journal version. Chen and Li
states in their paper that it was conjectured by Akira Saito, that $t \ge \left
\lceil \frac {2k} {3} \right \rceil$. They also states in their paper that they
believe $t \ge k - c$ for some constant $c$.
  In this note, we give a short proof to show that $t \ge \left \lceil \frac{3
k}{5}\right \rceil$, using an entirely different method. Our proof is only
about 2 pages long. The draw-back is that our bound is less by 1, than the
bound given by Chen and Li. We hope that the new approach adopted in this paper
would eventually lead to the settlement of the conjectures by Saito and/or Chen
and Li.
