{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:36Z","timestamp":1753893816212,"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>A 2016 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for $k \\ge 3$, if a graph has chromatic number greater than $k$, then it contains at least as many cycles of length $0 \\bmod k$ as the complete graph on $k+1$ vertices. Our main result confirms this in the $k=3$ case by showing every $4$-critical graph contains at least four cycles of length $0 \\bmod 3$, and that $K_4$ is the unique such graph achieving the minimum.\r\nWe make progress on the general conjecture as well, showing that $(k+1)$-critical graphs with minimum degree $k$ have at least as many cycles of length $0\\bmod r$ as $K_{k+1}$, provided $k+1 \\ne 0 \\bmod r$. We also show that $K_{k+1}$ uniquely minimizes the number of cycles of length $1\\bmod k$ among all $(k+1)$-critical graphs, strengthening a recent result of Moore and West and extending it to the $k=3$ case.<\/jats:p>","DOI":"10.37236\/12623","type":"journal-article","created":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T10:03:59Z","timestamp":1720778639000},"source":"Crossref","is-referenced-by-count":0,"title":["4-Chromatic Graphs Have At Least Four Cycles of Length $0 \\bmod 3$"],"prefix":"10.37236","volume":"31","author":[{"given":"Sean","family":"Kim","sequence":"first","affiliation":[]},{"given":"Michael","family":"Picollelli","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2024,6,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p58\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p58\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T10:03:59Z","timestamp":1720778639000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i2p58"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,28]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/12623","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,6,28]]},"article-number":"P2.58"}}