{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T14:14:59Z","timestamp":1772028899067,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to proper $k$-colourings of $G$ and edges between colourings that differ at a single vertex. The graph supports the Glauber dynamics Markov chain for $k$-colourings, and has been extensively studied from both extremal and probabilistic perspectives.\r\nIn this note, we show that for every graph $G$, there exists $k$ such that $G$ is uniquely determined by its $k$-colouring graph, confirming two conjectures of Asgarli, Krehbiel, Levinson and Russell. We further show that no finite family of generalised chromatic polynomials for $G$, which encode induced subgraph counts of its colouring graphs, uniquely determine $G$.<\/jats:p>","DOI":"10.37236\/12853","type":"journal-article","created":{"date-parts":[[2024,11,28]],"date-time":"2024-11-28T12:32:52Z","timestamp":1732797172000},"source":"Crossref","is-referenced-by-count":2,"title":["A Note on Graphs of $k$-Colourings"],"prefix":"10.37236","volume":"31","author":[{"given":"Emma","family":"Hogan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Scott","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Youri","family":"Tamitegama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jane","family":"Tan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,11,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i4p48\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i4p48\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,28]],"date-time":"2024-11-28T12:32:53Z","timestamp":1732797173000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i4p48"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,29]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,10,3]]}},"URL":"https:\/\/doi.org\/10.37236\/12853","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,29]]},"article-number":"P4.48"}}