{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:05:25Z","timestamp":1787321125359,"version":"build-2736575974"},"reference-count":9,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Abelian group colorings were first introduced by Jaeger et al. in [ J. Combin. Theory Ser. B, 56 (1992), pp. 165--182] as the dual concept of group connectivity of graphs. For given groups $\\Gamma_1$ and $\\Gamma_2$ with $|\\Gamma_1| = |\\Gamma_2|$, the dual version of a problem raised by Jaeger et al. suggests to investigate whether every $\\Gamma_1$-colorable graph $G$ is also $\\Gamma_2$-colorable. Recently, H\u01d4sek, Moheln\u00edkov\u00e1, and \u0160\u00e1mal [ J. Graph Theory, 93 (2019), pp. 317--327] used computer testing to find the first examples of $\\mathbb{Z}_4$-connected but not $\\mathbb{Z}_2^2$-connected graphs as well as $\\mathbb{Z}_2^2$-connected but not $\\mathbb{Z}_4$-connected graphs. As their examples are nonplanar, the group coloring problem remains unanswered. Group coloring was extended to non-abelian groups in Li and Lai [ Discrete Math., 313 (2013), pp. 101--104]. We introduce a group coloring local structure (defined as a snarl in the paper) and use it to construct infinitely many ordered triples $(G, \\Gamma_1, \\Gamma_2)$ in which $G$ is a graph and $\\Gamma_1$ and $\\Gamma_2$ are groups with $|\\Gamma_1| = |\\Gamma_2|$, such that $G$ is $\\Gamma_1$-colorable but not $\\Gamma_2$-colorable.<\/jats:p>","DOI":"10.1137\/19m1300546","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T08:53:49Z","timestamp":1635843229000},"page":"2535-2543","source":"Crossref","is-referenced-by-count":3,"title":["A Note on Group Colorings and Group Structure"],"prefix":"10.1137","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7698-2125","authenticated-orcid":true,"given":"Hong-Jian","family":"Lai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lucia","family":"Mazza","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,11,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-970-5"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1968.27.229"},{"key":"atypb3","unstructured":"T. W. Hungerford,\n                      Algebra\n                      , Springer, New York, 2000."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22488"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(92)90016-Q"},{"key":"atypb6","first-page":"405","volume":"27","author":"Lai H.-J.","year":"2011","journal-title":"Ser.)"},{"key":"atypb7","first-page":"299","volume":"62","author":"Lai H.-J.","year":"2002","journal-title":"Ars Combin."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2012.09.009"},{"key":"atypb9","volume-title":"An Introduction to the Theory of Groups","author":"Rotman J. J.","year":"1999"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1300546","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:06:32Z","timestamp":1787317592000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1300546"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/19M1300546"],"URL":"https:\/\/doi.org\/10.1137\/19m1300546","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}