{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T00:08:28Z","timestamp":1699834108940},"reference-count":7,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":8798,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1982,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let {<jats:italic>p<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>}<jats:italic>k<\/jats:italic>\u22652, <jats:italic>k<\/jats:italic>\u22604 be a sequence of non\u2010negative integers which satisfies 8 + \u03a3<jats:sub><jats:italic>k<\/jats:italic>\u22653<\/jats:sub>(<jats:italic>k<\/jats:italic> \u2014 4)<jats:italic>p<jats:sub>k<\/jats:sub><\/jats:italic> = 0. Then there exists an integer <jats:italic>p<\/jats:italic><jats:sub>4<\/jats:sub> such that there exists a 2\u2010connected planar graph with exactly <jats:italic>p<jats:sub>k<\/jats:sub><\/jats:italic> <jats:italic>k<\/jats:italic>\u2010gons as faces for all <jats:italic>k<\/jats:italic> \u2265 2. This paper determines all such <jats:italic>p<\/jats:italic><jats:sub>4<\/jats:sub> when <jats:italic>p<jats:sub>k<\/jats:sub><\/jats:italic> = 0 for <jats:italic>k<\/jats:italic> \u2265 5 and determines that there is a constant <jats:italic>C<\/jats:italic> \u2265 1 such that for some <jats:italic>m<\/jats:italic> \u2264 <jats:italic>p<jats:sub>2<\/jats:sub><\/jats:italic> + 1\/4<jats:italic>p<\/jats:italic><jats:sub>3<\/jats:sub> + <jats:italic>C<\/jats:italic>, there exists a 2\u2010connected planar graph with exactly <jats:italic>p<jats:sub>k<\/jats:sub><\/jats:italic> faces for each <jats:italic>p<\/jats:italic><jats:sub>4<\/jats:sub> = <jats:italic>m<\/jats:italic> + 2<jats:italic>w<\/jats:italic>, <jats:italic>w<\/jats:italic> a positive integer. When there exists at least one odd <jats:italic>k<\/jats:italic> \u2265 3 for which <jats:italic>p<jats:sub>k<\/jats:sub><\/jats:italic> \u2260 0, the coefficient 2 of <jats:italic>w<\/jats:italic> in the above equation may be replaced by 1. These conclusions do not hold if the coefficients of <jats:italic>p<\/jats:italic><jats:sub>2<\/jats:sub> and <jats:italic>p<\/jats:italic><jats:sub>3<\/jats:sub> are any smaller than 1 and 1\/4, respectively.<\/jats:p>","DOI":"10.1002\/jgt.3190060303","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T06:54:54Z","timestamp":1180421694000},"page":"255-281","source":"Crossref","is-referenced-by-count":3,"title":["4\u2010valent graphs"],"prefix":"10.1002","volume":"6","author":[{"given":"T. C.","family":"Enns","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(80)90229-0"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(74)80020-8"},{"key":"e_1_2_1_4_2","volume-title":"Convex Polytopes","author":"Gr\u00fcnbaum B.","year":"1967"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771220"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(74)90022-3"},{"key":"e_1_2_1_7_2","unstructured":"D. A.Roland An extension of Eberhard's theorem. M. Sci. Thesis University of Washington (1968)."},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758126"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190060303","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190060303","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,12]],"date-time":"2023-11-12T05:07:45Z","timestamp":1699765665000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190060303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1982,9]]}},"alternative-id":["10.1002\/jgt.3190060303"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190060303","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,9]]}}}