A colorful path in a graph $ G $ is a path with $\chi (G) $ vertices whose colors are different.
A $ v $-colorful path is such a path, starting from $ v $. Let $ G\neq C_7 $ be a connected
graph with maximum degree $\Delta (G) $. We show that there exists a $(\Delta (G)+ 1) $-
coloring of $ G $ with a $ v $-colorful path for every $ v\in V (G) $. We also prove that this
result is true if one replaces $(\Delta (G)+ 1) $ colors with $2\chi (G) $ colors. If $\chi
(G)=\omega (G) $, then the result still holds for $\chi (G) $ colors. For every graph $ G $, we …