Fan-goodness of sparse graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918091004313600 |
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| author | Huang, Ting Zhang, Yanbo Chen, Yaojun |
| author_facet | Huang, Ting Zhang, Yanbo Chen, Yaojun |
| contents | Let $G$ be a connected graph of order $n$, $F_k$ be a fan consisting of $k$ triangles sharing a common vertex, and $tF_k$ be $t$ vertex-disjoint copies of $F_k$. Brennan (2017) showed the Ramsey number $r(G,F_k)=2n-1$ for $G$ being a unicyclic graph for $n \geq k^2-k+1$ and $k\ge 18$, and asked the threshold $c(n)$ for which $r(G,F_k) \geq 2n$ holds for any $G$ containing at least $c(n)$ cycles and $n$ being large. In this paper, we consider fan-goodness of general sparse graphs and show that if $G$ has at most $n(1+ε(k))$ edges, where $ε(k)$ is a constant depending on $k$, then $$r(G,F_k)=2n-1$$ for $n\ge 36k^4$, which implies that $c(n)$ is greater than $ε(k) n$. Moreover, if $G$ has at most $n(1+ε(k,t))$ edges, where $ε(k,t)$ is a constant depending on $k,t$, then $$r(G,tF_k)=2n+t-2$$ provided $n\ge 161t^2k^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09832 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fan-goodness of sparse graphs Huang, Ting Zhang, Yanbo Chen, Yaojun Combinatorics 05C55, 05D10 Let $G$ be a connected graph of order $n$, $F_k$ be a fan consisting of $k$ triangles sharing a common vertex, and $tF_k$ be $t$ vertex-disjoint copies of $F_k$. Brennan (2017) showed the Ramsey number $r(G,F_k)=2n-1$ for $G$ being a unicyclic graph for $n \geq k^2-k+1$ and $k\ge 18$, and asked the threshold $c(n)$ for which $r(G,F_k) \geq 2n$ holds for any $G$ containing at least $c(n)$ cycles and $n$ being large. In this paper, we consider fan-goodness of general sparse graphs and show that if $G$ has at most $n(1+ε(k))$ edges, where $ε(k)$ is a constant depending on $k$, then $$r(G,F_k)=2n-1$$ for $n\ge 36k^4$, which implies that $c(n)$ is greater than $ε(k) n$. Moreover, if $G$ has at most $n(1+ε(k,t))$ edges, where $ε(k,t)$ is a constant depending on $k,t$, then $$r(G,tF_k)=2n+t-2$$ provided $n\ge 161t^2k^4$. |
| title | Fan-goodness of sparse graphs |
| topic | Combinatorics 05C55, 05D10 |
| url | https://arxiv.org/abs/2507.09832 |