Gallai-Ramsey Multiplicity

Publication date: 20 February 2023

Abstract: Given two graphs

G

and

H

, the emph{general

k

-colored Gallai-Ramsey number}

o p e r a t o r n a m e {g r}_{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 emph{Gallai-Ramsey multiplicity}

o p e r a t o r n a m e {G M}_{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_{o p e r a t o r n a m e {g r}_{k} (G, H)}

. In this paper, we give upper and lower bounds for Gallai-Ramsey multiplicity involving some small rainbow subgraphs.