{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T00:04:15Z","timestamp":1648771455239},"reference-count":0,"publisher":"Alliance of Diamond Open Access Journals","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AiC"],"abstract":"The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erd\u0151s from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollob\u00e1s in [Bull. London Math. Soc. 9 (1977), 97-98]( https:\/\/doi.org\/10.1112\/blms\/9.1.97). Another example is a problem of Erd\u0151s from the 1990s asking whether there exists $A\\subseteq\\mathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https:\/\/doi.org\/10.1002\/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs.\n\nThe current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.<\/jats:p>","DOI":"10.19086\/aic.18971","type":"journal-article","created":{"date-parts":[[2021,1,27]],"date-time":"2021-01-27T14:36:25Z","timestamp":1611758185000},"source":"Crossref","is-referenced-by-count":0,"title":["Cycle Lengths Modulo k in Large 3-connected Cubic Graphs, Advances in Combinatorics"],"prefix":"10.19086","author":[{"given":"Kasper S.","family":"Lyngsie","sequence":"first","affiliation":[{"name":"Technical University of Denmark"}]},{"ORCID":"http:\/\/orcid.org\/0000-0003-4555-6021","authenticated-orcid":false,"given":"Martin","family":"Merker","sequence":"additional","affiliation":[{"name":"Technical University of Denmark"}]}],"member":"8170","published-online":{"date-parts":[[2021,2,1]]},"container-title":["Advances in Combinatorics"],"original-title":[],"language":"en","deposited":{"date-parts":[[2021,1,27]],"date-time":"2021-01-27T14:36:26Z","timestamp":1611758186000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.advancesincombinatorics.com\/article\/18971-cycle-lengths-modulo-k-in-large-3-connected-cubic-graphs"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,1]]},"references-count":0,"URL":"http:\/\/dx.doi.org\/10.19086\/aic.18971","relation":{},"ISSN":["2517-5599"],"issn-type":[{"value":"2517-5599","type":"electronic"}],"subject":["Discrete Mathematics and Combinatorics"],"published":{"date-parts":[[2021,2,1]]}}}