The Ramsey numbers of squares of paths and cycles
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912486363496448 |
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| author | Allen, Peter Cecchelli, Domenico Mergoni Roberts, Barnaby Skokan, Jozef |
| author_facet | Allen, Peter Cecchelli, Domenico Mergoni Roberts, Barnaby Skokan, Jozef |
| contents | The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the edges of the complete graph $K_N$ are coloured with red and blue, there exists either a red copy of $G$ or a blue copy of $H$. We prove that for all sufficiently large $n$ we have \[R(P_{3n}^2,P_{3n}^2)=R(P_{3n+1}^2,P_{3n+1}^2)=R(C_{3n}^2,C_{3n}^2)=9n-3\mbox{ and } R(P_{3n+2}^2,P_{3n+2}^2)=9n+1.\] We also show that for any $γ>0$ and $Δ$ there exists $β>0$ such that the following holds. If $G$ can be coloured with three colours such that all colour classes have size at most $n$, the maximum degree $Δ(G)$ of $G$ is at most $Δ$, and $G$ has bandwidth at most $βn$, then $R(G,G)\le (3+γ)n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_14860 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Ramsey numbers of squares of paths and cycles Allen, Peter Cecchelli, Domenico Mergoni Roberts, Barnaby Skokan, Jozef Combinatorics The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the edges of the complete graph $K_N$ are coloured with red and blue, there exists either a red copy of $G$ or a blue copy of $H$. We prove that for all sufficiently large $n$ we have \[R(P_{3n}^2,P_{3n}^2)=R(P_{3n+1}^2,P_{3n+1}^2)=R(C_{3n}^2,C_{3n}^2)=9n-3\mbox{ and } R(P_{3n+2}^2,P_{3n+2}^2)=9n+1.\] We also show that for any $γ>0$ and $Δ$ there exists $β>0$ such that the following holds. If $G$ can be coloured with three colours such that all colour classes have size at most $n$, the maximum degree $Δ(G)$ of $G$ is at most $Δ$, and $G$ has bandwidth at most $βn$, then $R(G,G)\le (3+γ)n$. |
| title | The Ramsey numbers of squares of paths and cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2212.14860 |