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{\bf Saieed Akbari, Vahid Liaghat and Afshin Nikzad}
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{\bf Colorful Paths in Vertex Coloring of Graphs}
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A colorful path in a graph $G$ is a path with $\chi(G)$ vertices whose
colors are different. A \cpath{v} 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 \cpath{v} 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 $v \in V(G)$.


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