{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:18Z","timestamp":1753893798408,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $H$.\r\nWe prove that for all sufficiently large $n$ we have$$ R(P_{3n}^2,P_{3n}^2)=R(P_{3n+1}^2,P_{3n+1}^2)=R(C_{3n}^2,C_{3n}^2)=9n-3\\mbox{ and } R(P_{3n+2}^2,P_{3n+2}^2)=9n+1.$$\r\nWe also show that for any $\\gamma&gt;0$ and $\\Delta$ there exists $\\beta&gt;0$ 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 $R(G,G)\\le (3+\\gamma)n$.<\/jats:p>","DOI":"10.37236\/11847","type":"journal-article","created":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T08:18:02Z","timestamp":1713428282000},"source":"Crossref","is-referenced-by-count":0,"title":["The Ramsey Numbers of Squares of Paths and Cycles"],"prefix":"10.37236","volume":"31","author":[{"given":"Domenico","family":"Mergoni Cecchelli","sequence":"first","affiliation":[]},{"given":"Peter","family":"Allen","sequence":"additional","affiliation":[]},{"given":"Jozef","family":"Skokan","sequence":"additional","affiliation":[]},{"given":"Barnaby","family":"Roberts","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2024,4,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p11\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p11\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T08:18:02Z","timestamp":1713428282000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i2p11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,5]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/11847","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,4,5]]},"article-number":"P2.11"}}