| Summary: | 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 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 S3+S_3^ + be the graph on four vertices consisting of a triangle with a pendant edge. In this paper, we prove that grk(S3+:P5)=k+4(k≥5)g{r_k}\left( {S_3^ + :{P_5}} \right) = k + 4\left( {k \ge 5} \right), grk(S3+:mP2)=(m-1)k+m+1(k≥1)g{r_k}\left( {S_3^ + :m{P_2}} \right) = \left( {m - 1} \right)k + m + 1\left( {k \ge 1} \right), grk(S3+:P3∪P2)=k+4(k≥5)g{r_k}\left( {S_3^ + :{P_3} \cup {P_2}} \right) = k + 4\left( {k \ge 5} \right) and grk(S3+:2P3)=k+5(k≥1)g{r_k}\left( {S_3^ + :2{P_3}} \right) = k + 5\left( {k \ge 1} \right).
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