  Let $G$ be an edge-colored graph. A heterochromatic (rainbow, or
multicolored) path of $G$ is such a path in which no two edges have the same
color. Let $d^c(v)$ denote the color degree and $CN(v)$ denote the color
neighborhood of a vertex $v$ of $G$. In a previous paper, we showed that if
$d^c(v)\geq k$ (color degree condition) for every vertex $v$ of $G$, then $G$
has a heterochromatic path of length at least $\lceil\frac{k+1}{2}\rceil$, and
if $|CN(u)\cup CN(v)|\geq s$ (color neighborhood union condition) for every
pair of vertices $u$ and $v$ of $G$, then $G$ has a heterochromatic path of
length at least $\lceil\frac{s}{3}\rceil+1$. Later, in another paper we first
showed that if $k\leq 7$, $G$ has a heterochromatic path of length at least
$k-1$, and then, based on this we use induction on $k$ and showed that if
$k\geq 8$, then $G$ has a heterochromatic path of length at least
$\lceil\frac{3k}{5}\rceil+1$. In the present paper, by using a simpler approach
we further improve the result by showing that if $k\geq 8$, $G$ has a
heterochromatic path of length at least $\lceil\frac{2k}{3}\rceil+1$, which
confirms a conjecture by Saito. We also improve a previous result by showing
that under the color neighborhood union condition, $G$ has a heterochromatic
path of length at least $\lfloor\frac{2s+4}{5}\rfloor$.
