Gallai-Ramsey numbers for rainbow S+3 and monochromatic paths

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 GallaiRamsey number grk(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 grk(S + 3 : P5) = k+ 4 (k ≥ 5), grk(S + 3 : mP2) = (m− 1)k+m+ 1 (k ≥ 1), grk(S 3 : P3 ∪ P2) = k + 4 (k ≥ 5) and grk(S + 3 : 2P3) = k + 5 (k ≥ 1).