{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:41:33Z","timestamp":1757619693635,"version":"3.44.0"},"reference-count":8,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:00:00Z","timestamp":1753401600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:00:00Z","timestamp":1753401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100008047","name":"Carnegie Mellon University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100008047","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We say a graph <jats:italic>H<\/jats:italic> decomposes a graph <jats:italic>G<\/jats:italic> if there exists a partition of the edges of <jats:italic>G<\/jats:italic> into subgraphs isomorphic to <jats:italic>H<\/jats:italic>. We seek to characterize necessary and sufficient conditions for a cycle of length <jats:italic>k<\/jats:italic>, denoted <jats:inline-formula>\n              <jats:tex-math>$$C_k$$<\/jats:tex-math>\n            <\/jats:inline-formula>, to decompose the Cartesian product of two cycles <jats:inline-formula>\n              <jats:tex-math>$$C_m ~\\square ~ C_n$$<\/jats:tex-math>\n            <\/jats:inline-formula>. We prove that if <jats:italic>m<\/jats:italic> is a multiple of 3, then the Cartesian product of a cycle <jats:inline-formula>\n              <jats:tex-math>$$C_m$$<\/jats:tex-math>\n            <\/jats:inline-formula> and any other cycle can be decomposed into 3 cycles of equal length. This extends work of Kotzig, who proved in 1973 that the Cartesian product of two cycles can always be decomposed into two cycles of equal length. We also show that if <jats:italic>k<\/jats:italic>, <jats:italic>m<\/jats:italic>, and <jats:italic>n<\/jats:italic> are positive, and <jats:italic>k<\/jats:italic> divides 4<jats:italic>mn<\/jats:italic>, then <jats:inline-formula>\n              <jats:tex-math>$$C_{4k}$$<\/jats:tex-math>\n            <\/jats:inline-formula> decomposes <jats:inline-formula>\n              <jats:tex-math>$$C_{4m} ~\\square ~ C_{4n}$$<\/jats:tex-math>\n            <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00373-025-02953-2","type":"journal-article","created":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T06:50:37Z","timestamp":1753426237000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Cycle Decompositions of Cartesian Products of Two Cycles"],"prefix":"10.1007","volume":"41","author":[{"given":"Moriah","family":"Aberle","sequence":"first","affiliation":[]},{"given":"Sarah","family":"Gold","sequence":"additional","affiliation":[]},{"given":"Rivkah","family":"Moshe","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3413-7991","authenticated-orcid":false,"given":"David","family":"Offner","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,25]]},"reference":[{"key":"2953_CR1","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/978-94-009-0517-7_2","volume-title":"Cycles and Rays","author":"B Alspach","year":"1990","unstructured":"Alspach, B., Bermond, J.-C., Sotteau, D.: Decomposition into cycles I: Hamilton decompositions. In: Hahn, G., et al. (eds.) Cycles and Rays, pp. 9\u201318. Kluwer Academic, Dordrecht (1990)"},{"key":"2953_CR2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2021.103320","volume":"95","author":"M Axenovich","year":"2021","unstructured":"Axenovich, M., Offner, D., Tompkins, C.: Long path and cycle decompositions of even hypercubes. Eur. J. Combin. 95, 103320 (2021)","journal-title":"Eur. J. Combin."},{"issue":"2","key":"2953_CR3","doi-asserted-by":"publisher","first-page":"233","DOI":"10.2140\/involve.2024.17.233","volume":"17","author":"S Gibson","year":"2024","unstructured":"Gibson, S., Offner, D.: Decompositions of even hypercubes into cycles whose length is a power of two. Involve 17(2), 233\u2013247 (2024)","journal-title":"Involve"},{"issue":"1","key":"2953_CR4","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1017\/S1446788700000112","volume":"61","author":"P Horak","year":"1996","unstructured":"Horak, P., Siran, J., Wallis, W.: Decomposing cubes. J. Aust. Math. Soc. Ser. A 61(1), 119\u2013128 (1996)","journal-title":"J. Aust. Math. Soc. Ser. A"},{"key":"2953_CR5","volume-title":"Every Cartesian Product of Two Circuits is Decomposable into Two Hamiltonian Circuits","author":"A Kotzig","year":"1973","unstructured":"Kotzig, A.: Every Cartesian Product of Two Circuits is Decomposable into Two Hamiltonian Circuits. Centre de Recherches Mathematique, Montreal (1973)"},{"issue":"3","key":"2953_CR6","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1007\/s00373-013-1402-0","volume":"31","author":"M Mollard","year":"2015","unstructured":"Mollard, M., Ramras, M.: Edge decompositions of hypercubes by paths and by cycles. Graphs Combin. 31(3), 729\u2013741 (2015)","journal-title":"Graphs Combin."},{"issue":"2","key":"2953_CR7","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1016\/0012-365X(91)90354-5","volume":"90","author":"R Stong","year":"1991","unstructured":"Stong, R.: Hamilton decompositions of Cartesian products of graphs. Discrete Math. 90(2), 169\u2013190 (1991)","journal-title":"Discrete Math."},{"key":"2953_CR8","first-page":"443","volume":"74","author":"S Tapadia","year":"2019","unstructured":"Tapadia, S., Waphare, B., Borse, Y.: Cycle decompositions of the Cartesian product of cycles. Australas. J. Combin. 74, 443\u2013459 (2019)","journal-title":"Australas. J. Combin."}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-025-02953-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-025-02953-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-025-02953-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,7]],"date-time":"2025-09-07T22:52:03Z","timestamp":1757285523000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-025-02953-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,25]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["2953"],"URL":"https:\/\/doi.org\/10.1007\/s00373-025-02953-2","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"type":"print","value":"0911-0119"},{"type":"electronic","value":"1435-5914"}],"subject":[],"published":{"date-parts":[[2025,7,25]]},"assertion":[{"value":"19 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"84"}}