the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...

The problem of finding paths in temporal graphs has been recently considered due to its many applications. In this paper we consider a variant the that, given vertex-colored graph, asks for path whose vertices have distinct colors and include maximum number colors. We study approximation complexity provide an inapproximability lower bound. Then present heuristic experimental evaluation our heur...

Let G be a connected graph of chromatic number k. For a k-coloring f of G, a full f -rainbow path is a path of order k in G whose vertices are all colored differently by f . We show that G has a k-coloring f such that every vertex of G lies on a full f -rainbow path, which provides a positive answer to a question posed by Lin (Simple proofs of results on paths representing all colors in proper ...

Kierstead, H.A. and W.T. Trotter. Colorful induced subgraphs, Discrete Mathematics 101 (1992) 165-169. A colored graph is a graph whose vertices have been properly, though not necessarily optimally colored, with integers. Colored graphs have a natural orientation in which edges are directed from the end point with smaller color to the end point with larger color. A subgraph of a colored graph i...

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...

In this paper, we prove that every 3-chromatic connected graph, except C7, admits a 3-vertex coloring in which every vertex is the beginning of a 3-chromatic path. It is a special case of a conjecture due to S. Akbari, F. Khaghanpoor, and S. Moazzeni, cited in [P.J. Cameron, Research problems from the BCC22, Discrete Math. 311 (2011), 1074–1083], stating that every connected graph G other than ...