{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T15:00:18Z","timestamp":1781276418570,"version":"3.54.1"},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T00:00:00Z","timestamp":1777939200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T00:00:00Z","timestamp":1777939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>A graph is said to be neighborhood 3-balanced if there exists a vertex labeling with three colors so that each vertex has an equal number of neighbors of each color. We give order constraints on 3-balanced graphs, determine which generalized Petersen and Pappus graphs are 3-balanced, discuss when being 3-balanced is preserved under various graph constructions, give two general characterizations of cubic 3-balanced graphs, and classify cubic 3-balanced graphs of small order.<\/jats:p>","DOI":"10.1007\/s00373-026-03038-4","type":"journal-article","created":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T17:07:50Z","timestamp":1778000870000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neighborhood 3-Balanced Graphs"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-4676-6948","authenticated-orcid":false,"given":"Mitchell","family":"Minyard","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark R.","family":"Sepanski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,5]]},"reference":[{"issue":"1","key":"3038_CR1","first-page":"45","volume":"1","author":"H Aram","year":"2012","unstructured":"Aram, H., Sheikholeslami, S.M., Volkmann, L.: On the total domatic number of regular graphs. Trans. Comb. 1(1), 45\u201351 (2012). (ISSN 2251-8657,2251-8665)","journal-title":"Trans. Comb."},{"issue":"1","key":"3038_CR2","doi-asserted-by":"publisher","first-page":"345","DOI":"10.5614\/ejgta.2022.10.1.25","volume":"10","author":"S Biswas","year":"2022","unstructured":"Biswas, S., Das, A.: A generalization of Pappus graph. Electron. J. Graph Theory Appl. (EJGTA) 10(1), 345\u2013356 (2022). https:\/\/doi.org\/10.5614\/ejgta.2022.10.1.25. (ISSN 2338-2287.)","journal-title":"Electron. J. Graph Theory Appl. (EJGTA)"},{"issue":"2\u20133","key":"3038_CR3","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1016\/0095-8956(77)90034-x","volume":"23","author":"FC Bussemaker","year":"1977","unstructured":"Bussemaker, F.C., \u010cobelji\u0107, S., Cvetkovi\u0107, D.M., Seidel, J.J.: Cubic graphs on $$\\le 14$$ vertices. J. Combinatorial Theory Ser. B 23(2\u20133), 234\u2013235 (1977). https:\/\/doi.org\/10.1016\/0095-8956(77)90034-x. (ISSN 0095-8956.)","journal-title":"J. Combinatorial Theory Ser. B"},{"key":"3038_CR4","doi-asserted-by":"crossref","unstructured":"Castagna, F., Prins, G.: Every generalized Petersen graph has a Tait coloring. Pacific J. Math., 40: 53\u201358, (1972). ISSN 0030-8730,1945-5844. http:\/\/projecteuclid.org\/euclid.pjm\/1102968820","DOI":"10.2140\/pjm.1972.40.53"},{"key":"3038_CR5","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/j.dam.2015.04.026","volume":"193","author":"B Chen","year":"2015","unstructured":"Chen, B., Kim, J.H., Tait, M., Verstraete, J.: On coupon colorings of graphs. Discrete Appl. Math. 193, 94\u2013101 (2015). https:\/\/doi.org\/10.1016\/j.dam.2015.04.026. (ISSN 0166-218X,1872-6771)","journal-title":"Discrete Appl. Math."},{"key":"3038_CR6","first-page":"247","volume":"73","author":"P Dankelmann","year":"2004","unstructured":"Dankelmann, P., Calkin, N.: The domatic number of regular graphs. Ars Combin. 73, 247\u2013255 (2004). (ISSN 0381-7032,2817-5204)","journal-title":"Ars Combin."},{"key":"3038_CR7","doi-asserted-by":"publisher","unstructured":"Freyberg, B., Marr, A.: Neighborhood balanced colorings of graphs. Graphs Combin., 40 (2): Paper No. 41, 13, 2024. ISSN 0911-0119,1435-5914. https:\/\/doi.org\/10.1007\/s00373-024-02766-9","DOI":"10.1007\/s00373-024-02766-9"},{"issue":"1\u20132","key":"3038_CR8","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/S0012-365X(01)00466-6","volume":"256","author":"G Hahn","year":"2002","unstructured":"Hahn, G., Kratochv\u00edl, J., \u0160ir\u00e1\u0148, J., Sotteau, D.: On the injective chromatic number of graphs. Discrete Math. 256(1\u20132), 179\u2013192 (2002). https:\/\/doi.org\/10.1016\/S0012-365X(01)00466-6. (ISSN 0012-365X,1872-681X.)","journal-title":"Discrete Math."},{"key":"3038_CR9","doi-asserted-by":"publisher","first-page":"221","DOI":"10.2307\/2319844","volume":"82","author":"R Isaacs","year":"1975","unstructured":"Isaacs, R.: Infinite families of nontrivial trivalent graphs which are not Tait colorable. Amer. Math. Monthly 82, 221\u2013239 (1975). https:\/\/doi.org\/10.2307\/2319844. (ISSN 0002-9890,1930-0972.)","journal-title":"Amer. Math. Monthly"},{"issue":"1","key":"3038_CR10","first-page":"1","volume":"46","author":"Z Li","year":"2020","unstructured":"Li, Z., Shao, Z., Zhu, E.: Injective coloring of generalized Petersen graphs. Houston J. Math. 46(1), 1\u201312 (2020). (ISSN 0362-1588)","journal-title":"Houston J. Math."},{"key":"3038_CR11","doi-asserted-by":"publisher","unstructured":"Mann, H.B.: On linear relations between roots of unity. Mathematika 12, 107\u2013117 (1965). https:\/\/doi.org\/10.1112\/S0025579300005210. (ISSN 0025-5793.)","DOI":"10.1112\/S0025579300005210"},{"key":"3038_CR12","doi-asserted-by":"publisher","unstructured":"Meringer, M.: Fast generation of regular graphs and construction of cages. J. Graph Theory 30(2), 137\u2013146 (1999). https:\/\/doi.org\/10.1002\/(SICI)1097-0118(199902)30:2%3C137::AID-JGT7%3E3.0.CO;2-G. (ISSN 0364-9024,1097-0118.)","DOI":"10.1002\/(SICI)1097-0118(199902)30:2%3C137::AID-JGT7%3E3.0.CO;2-G"},{"key":"3038_CR13","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1016\/S0021-9800(69)80116-X","volume":"6","author":"ME Watkins","year":"1969","unstructured":"Watkins, M.E.: A theorem on Tait colorings with an application to the generalized Petersen graphs. J. Combinatorial Theory 6, 152\u2013164 (1969). (ISSN 0021-9800)","journal-title":"J. Combinatorial Theory"}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-026-03038-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-026-03038-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-026-03038-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T14:26:27Z","timestamp":1781274387000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-026-03038-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,5]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["3038"],"URL":"https:\/\/doi.org\/10.1007\/s00373-026-03038-4","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"value":"0911-0119","type":"print"},{"value":"1435-5914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,5]]},"assertion":[{"value":"25 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2026","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 relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"50"}}