Multicolor Ramsey Number for Double Stars
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866911438740652032 |
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| author | Ruotolo, Jake Song, Zi-Xia |
| author_facet | Ruotolo, Jake Song, Zi-Xia |
| contents | For a graph $H$ and an integer $k\ge1$, let $r(H;k)$ and $r_\ell(H;k)$ denote the $k$-color Ramsey number and list Ramsey number of $H$, respectively. Alon, Bucić, Kalvari, Kuperwasser and Szabó in 2021 initiated the systematic study of list Ramsey numbers of graphs and hypergraphs, and conjectured that $ r(K_{1,n};k)$ and $r_\ell(K_{1,n};k)$ are always equal. Motivated by their work, we study the $k$-color Ramsey number for double stars $S(n,m)$, where $n\ge m\ge1$. To the best of our knowledge, little is known on the exact value of $r(S(n,m);k)$ when $k\ge3$. A classic result of Erdős and Graham from 1975 asserts that $r(T;k)>k(n-1)+1$ for every tree $T$ with $n\ge 1$ edges and $k$ sufficiently large such that $n$ divides $k-1$. Using a folklore double counting argument in set system and the edge chromatic number of complete graphs, we prove that if $k$ is odd and $n$ is sufficiently large compared with $m$ and $k$, then \[ r(S(n,m);k)=kn+m+2.\] This is a step in our effort to determine whether $r(S(n,m);k)$ and $r_\ell(S(n,m);k)$ are always equal, which remains wide open. We also prove that $ r(S^m_n;k)=k(n-1)+m+2$ if $k $ is odd and $n$ is sufficiently large compared with $m$ and $k$, where $1\le m\le n$ and $S^m_n$ is obtained from $K_{1, n}$ by subdividing $m$ edges each exactly once. We end the paper with some observations towards the list Ramsey number for $S(n,m)$ and $S^m_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_03642 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multicolor Ramsey Number for Double Stars Ruotolo, Jake Song, Zi-Xia Combinatorics 05C55 For a graph $H$ and an integer $k\ge1$, let $r(H;k)$ and $r_\ell(H;k)$ denote the $k$-color Ramsey number and list Ramsey number of $H$, respectively. Alon, Bucić, Kalvari, Kuperwasser and Szabó in 2021 initiated the systematic study of list Ramsey numbers of graphs and hypergraphs, and conjectured that $ r(K_{1,n};k)$ and $r_\ell(K_{1,n};k)$ are always equal. Motivated by their work, we study the $k$-color Ramsey number for double stars $S(n,m)$, where $n\ge m\ge1$. To the best of our knowledge, little is known on the exact value of $r(S(n,m);k)$ when $k\ge3$. A classic result of Erdős and Graham from 1975 asserts that $r(T;k)>k(n-1)+1$ for every tree $T$ with $n\ge 1$ edges and $k$ sufficiently large such that $n$ divides $k-1$. Using a folklore double counting argument in set system and the edge chromatic number of complete graphs, we prove that if $k$ is odd and $n$ is sufficiently large compared with $m$ and $k$, then \[ r(S(n,m);k)=kn+m+2.\] This is a step in our effort to determine whether $r(S(n,m);k)$ and $r_\ell(S(n,m);k)$ are always equal, which remains wide open. We also prove that $ r(S^m_n;k)=k(n-1)+m+2$ if $k $ is odd and $n$ is sufficiently large compared with $m$ and $k$, where $1\le m\le n$ and $S^m_n$ is obtained from $K_{1, n}$ by subdividing $m$ edges each exactly once. We end the paper with some observations towards the list Ramsey number for $S(n,m)$ and $S^m_n$. |
| title | Multicolor Ramsey Number for Double Stars |
| topic | Combinatorics 05C55 |
| url | https://arxiv.org/abs/2211.03642 |