Abstract
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.
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Acknowledgements
We thank Stéphane Bessy for valuable discussion on this topic.
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Bendele, O., Rautenbach, D. Optimal Colorings with Rainbow Paths. Graphs and Combinatorics 33, 729–734 (2017). https://doi.org/10.1007/s00373-017-1808-1
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DOI: https://doi.org/10.1007/s00373-017-1808-1