Gallai–Ramsey Multiplicity

Given two graphs G and H , the general k -colored Gallai–Ramsey number 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 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 gr k ( G : H ) . In this paper, we give upper and lower bounds for Gallai–Ramsey multiplicity involving some small rainbow subgraphs.