Colorful Paths in Vertex Coloring of Graphs

A colorful path in a graph G is a path with χ(G) vertices whose colors are different. A vcolorful path is such a path, starting from v. Let G 6= C7 be a connected graph with maximum degree ∆(G). We show that there exists a (∆(G) + 1)-coloring of G with a v-colorful path for every v ∈ V (G). We also prove that this result is true if one replaces (∆(G) + 1) colors with 2χ(G) colors. If χ(G) = ω(G), then the result still holds for χ(G) colors. For every graph G, we show that there exists a χ(G)-coloring of G with a rainbow path of length ⌊χ(G)/2⌋ starting from each v ∈ V (G).