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- W4313447745 abstract "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-3mbox{ and } R(P_{3n+2}^2,P_{3n+2}^2)=9n+1.] We also show that for any $gamma>0$ and $Delta$ there exists $beta>0$ 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 $Delta(G)$ of $G$ is at most $Delta$, and $G$ has bandwidth at most $beta n$, then $R(G,G)le (3+gamma)n$." @default.
- W4313447745 created "2023-01-06" @default.
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- W4313447745 date "2022-12-30" @default.
- W4313447745 modified "2023-10-18" @default.
- W4313447745 title "The Ramsey numbers of squares of paths and cycles" @default.
- W4313447745 doi "https://doi.org/10.48550/arxiv.2212.14860" @default.
- W4313447745 hasPublicationYear "2022" @default.
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