Ramsey and Gallai-Ramsey numbers for multiple triangles of graphs and their multiplicities

Given two graphs G and H, the k-edge-colored Gallai-Ramsey number grk(G : H) is defined to be minimum integer n such that every k-edge-coloring of the complete graph Kn contains either a rainbow copy of G or a monochromatic copy of H. The Gallai-Ramsey multiplicity GMk(G, H) is defined as the minimum total number of rainbow copy of G and monochromatic copy of H in any exact k-edge-coloring of Kgrk(G,H). In this paper, we determine the exact values of Gallai-Ramsey number grk(G : H) and Ramsey numbers R2(H), where H denotes the union of matchings and triangles, and G denotes a 3-star or a 4-path. We give the exact values for Ramsey multiplicity involving multiple matchings and triangle, and get the upper and lower bounds for Gallai-Ramsey multiplicity involving some small rainbow subgraphs. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.