Gallai-Ramsey multiplicity for rainbow small trees
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913280477364224 |
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| author | Li, Xueliang Si, Yuan |
| author_facet | Li, Xueliang Si, Yuan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_08370 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gallai-Ramsey multiplicity for rainbow small trees Li, Xueliang Si, Yuan Combinatorics 05C15 05C30 05C55 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. |
| title | Gallai-Ramsey multiplicity for rainbow small trees |
| topic | Combinatorics 05C15 05C30 05C55 |
| url | https://arxiv.org/abs/2309.08370 |