Geodesic
Gallai-Ramsey Numbers for Rainbow $S_3^+$ and Monochromatic Paths
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 349-362

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Motivated by Ramsey theory and other rainbow-coloring-related problems, we consider edge-colorings of complete graphs without rainbow copy of some fixed subgraphs. Given two graphs G and H, the k-colored Gallai-Ramsey number gr_k(G : H) is defined to be the minimum positive integer n such that every k-coloring of the complete graph on n vertices contains either a rainbow copy of G or a monochromatic copy of H. Let S_3^+ be the graph on four vertices consisting of a triangle with a pendant edge. In this paper, we prove that gr_k(S_3^+ : P_5) = k+4 (k ≥ 5), gr_k(S_3^+ : mP_2) = (m-1)k+m+1 (k ≥ 1), gr_k(S_3^+ : P_3 ∪ P_2) = k+4 (k ≥ 5) and gr_k( S_3^+ : 2P_3) = k+5 (k ≥1).
Keywords: Gallai-Ramsey number, rainbow coloring, monochromatic paths 
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     author = {Li, Xihe and Wang, Ligong},
     title = {Gallai-Ramsey {Numbers} for {Rainbow} $S_3^+$  and {Monochromatic} {Paths}},
     journal = {Discussiones Mathematicae. Graph Theory},
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     year = {2022},
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     url = {http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a2/}
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Li, Xihe; Wang, Ligong. Gallai-Ramsey Numbers for Rainbow $S_3^+$  and Monochromatic Paths. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 2, pp. 349-362. http://geodesic.mathdoc.fr/item/DMGT_2022_42_2_a2/