  In this paper, graphs under consideration are always edge-colored. We
consider long heterochromatic paths in heterochromatic triangle free graphs.
Two kinds of such graphs are considered, one is complete graphs with Gallai
colorings, i.e., heterochromatic triangle free complete graphs; the other is
heterochromatic triangle free graphs with $k$-good colorings, i.e., minimum
color degree at least $k$. For the heterochromatic triangle free graphs $K_n$,
we obtain that for every vertex $v\in V(K_n)$, $K_n$ has a heterochromatic
$v$-path of length at least $d^c(v)$; whereas for the heterochromatic triangle
free graphs $G$ we show that if, for any vertex $v\in V(G)$, $d^c(v)\geq k\geq
6$, then $G$ a heterochromatic path of length at least $\frac{3k}{4}$.
