Size Ramsey numbers of small graphs versus fans or paths
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2024
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| _version_ | 1866916196224335872 |
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| author | Li, Yufan Zhang, Yanbo Zhang, Yunqing |
| author_facet | Li, Yufan Zhang, Yanbo Zhang, Yunqing |
| contents | For two graphs $G_1$ and $G_2$, the size Ramsey number $\hat{r}(G_1,G_2)$ is the smallest positive integer $m$ for which there exists a graph $G$ of size $m$ such that for any red-blue edge-coloring of the graph $G$, $G$ contains either a red subgraph isomorphic to $G_1$, or a blue subgraph isomorphic to $G_2$. Let $P_n$ be a path with $n$ vertices, $nK_2$ a matching with $n$ edges, and $F_n$ a graph with $n$ triangles sharing exactly one vertex. If $G_1$ is a small fixed graph and $G_2$ denotes any graph from a graph class, one can sometimes completely determine $\hat{r}(G_1,G_2)$. Faudree and Sheehan confirmed all size Ramsey numbers of $P_3$ versus complete graphs in 1983. The next year Erdős and Faudree confirmed that of $2K_2$ versus complete graphs and complete bipartite graphs. We obtain three more Ramsey results of this type. For $n\ge 3$, we prove that $\hat{r}(P_3,F_n)=4n+4$ if $n$ is odd, and $\hat{r}(P_3,F_n)=4n+5$ if $n$ is even. This result refutes a conjecture proposed by Baskoro et al. We also show that $\hat{r}(2K_2,F_2)=12$ and $\hat{r}(2K_2,F_n)=5n+3$ for $n\ge 3$. In addition, we prove that $\hat{r}(2K_2,nP_m)=\min\{nm+1, (n+1)(m-1)\}$. This result verifies a conjecture posed by Vito and Silaban. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_04432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Size Ramsey numbers of small graphs versus fans or paths Li, Yufan Zhang, Yanbo Zhang, Yunqing Combinatorics 05C55, 05D10 For two graphs $G_1$ and $G_2$, the size Ramsey number $\hat{r}(G_1,G_2)$ is the smallest positive integer $m$ for which there exists a graph $G$ of size $m$ such that for any red-blue edge-coloring of the graph $G$, $G$ contains either a red subgraph isomorphic to $G_1$, or a blue subgraph isomorphic to $G_2$. Let $P_n$ be a path with $n$ vertices, $nK_2$ a matching with $n$ edges, and $F_n$ a graph with $n$ triangles sharing exactly one vertex. If $G_1$ is a small fixed graph and $G_2$ denotes any graph from a graph class, one can sometimes completely determine $\hat{r}(G_1,G_2)$. Faudree and Sheehan confirmed all size Ramsey numbers of $P_3$ versus complete graphs in 1983. The next year Erdős and Faudree confirmed that of $2K_2$ versus complete graphs and complete bipartite graphs. We obtain three more Ramsey results of this type. For $n\ge 3$, we prove that $\hat{r}(P_3,F_n)=4n+4$ if $n$ is odd, and $\hat{r}(P_3,F_n)=4n+5$ if $n$ is even. This result refutes a conjecture proposed by Baskoro et al. We also show that $\hat{r}(2K_2,F_2)=12$ and $\hat{r}(2K_2,F_n)=5n+3$ for $n\ge 3$. In addition, we prove that $\hat{r}(2K_2,nP_m)=\min\{nm+1, (n+1)(m-1)\}$. This result verifies a conjecture posed by Vito and Silaban. |
| title | Size Ramsey numbers of small graphs versus fans or paths |
| topic | Combinatorics 05C55, 05D10 |
| url | https://arxiv.org/abs/2404.04432 |