Optimal Colorings with Rainbow Paths

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 vertex-colorings, Graphs Combin. 23 2007 , 201–203). Furthermore, we show that if G has a cycle of length 0 modulo k , then G has a k -coloring f such that, for every vertex u of G , some full f -rainbow path begins at u , which solves a problem posed by Bessy and Bousquet (Colorful paths for 3-chromatic graphs, arXiv:1503.00965v1 ) and verifies some special cases of a conjecture of Akbari et al. (Colorful paths in vertex coloring of graphs, preprint). Finally, we establish some more results on the existence of optimal colorings with (directed) full rainbow paths.