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 not equal 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 epsilon 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 left perpendicular chi(G)/2right perpendicular starting from each v epsilon V(G).