Colorful Paths in Vertex Coloring of Graphs
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 show that there exists a $\chi(G)$-coloring of $G$ with a rainbow path of length $\lfloor\chi(G)/2\rfloor$ starting from each $v \in V(G)$.