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arXiv · 2309.08370

Gallai-Ramsey multiplicity for rainbow small trees

Xueliang Li·Yuan Si
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Abstract

Let $G, H$ be two non-empty graphs and $k$ be a positive integer. The Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum positive integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains either a rainbow subgraph $G$ or a monochromatic subgraph $H$. The Gallai-Ramsey multiplicity $\operatorname{GM}_k(G:H)$ is defined as the minimum total number of rainbow subgraphs $G$ and monochromatic subgraphs $H$ for all $k$-edge-colored $K_{\operatorname{gr}_k(G:H)}$. In this paper, we get some exact values of the Gallai-Ramsey multiplicity for rainbow small trees versus general monochromatic graphs under a sufficiently large number of colors. We also study the bipartite Gallai-Ramsey multiplicity.

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Xueliang Li, Yuan Si. 2023-09-15. Gallai-Ramsey multiplicity for rainbow small trees. https://arxiv.org/abs/2309.08370

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