{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:19Z","timestamp":1753893799220,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Extending Gr\u00f6tzsch's 3-coloring theorem in the flow setting, Steinberg and Younger in 1989 proved that every 4-edge-connected planar or projective planar graph admits a nowhere-zero 3-flow (3-NZF for short), while Tutte's 3-flow conjecture asserts all 4-edge-connected graphs admit 3-NZFs. In this paper, we generalize Gr\u00f6tzsch's theorem to signed planar graphs by showing that every 4-edge-connected signed planar graph with two negative edges admits a 3-NZF. On the other hand, a result from M\u00e1\u010dajov\u00e1 and \u0160koviera implies that there exist infinitely many 4-edge-connected signed planar graphs with three negative edges admitting no 3-NZFs but permitting 4-NZFs. Our proof employs the flow extension ideas from Steinberg-Younger and Thomassen, as well as refined exploration of the location of negative edges and elaborated discharging arguments in signed planar graphs.<\/jats:p>","DOI":"10.37236\/11892","type":"journal-article","created":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T16:38:31Z","timestamp":1718296711000},"source":"Crossref","is-referenced-by-count":0,"title":["Nowhere-Zero 3-Flows in Signed Planar Graphs"],"prefix":"10.37236","volume":"31","author":[{"given":"Jiaao","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yulai","family":"Ma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhengke","family":"Miao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongtang","family":"Shi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weifan","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,6,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T16:38:31Z","timestamp":1718296711000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i2p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,14]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/11892","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,6,14]]},"article-number":"P2.47"}}