Given two graphs G and H, the general k-colored Gallai–Ramsey number \({\text {gr}}_k(G:H)\) is defined to be the minimum integer m such that every k-coloring of the complete graph on m vertices contains either a rainbow copy of G or a monochromatic copy of H. Interesting problems arise when one asks how many such rainbow copy of G and monochromatic copy of H must occur. The Gallai–Ramsey multiplicity \({\text {GM}}_{k}(G:H)\) is defined as the minimum total number of rainbow copy of G and monochromatic copy of H in any exact k-coloring of \(K_{{\text {gr}}_{k}(G:H)}\) . In this paper, we give upper and lower bounds for Gallai–Ramsey multiplicity involving some small rainbow subgraphs.
