{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:34Z","timestamp":1753893814431,"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>In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere-zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4. Jaeger conjectured a generalization of Tutte's nowhere-zero 3-flow conjecture, namely, that every $(4p+1)$-regular, $4p$-edge-connected graph has an edge orientation in which every in-degree is either $p$ or $3p+1$. Inspired by the work of Pra\u0142at and Wormald investigating $p=1$, we address $p=2$ to show that the conjecture holds asymptotically almost surely for random 9-regular graphs. It follows that the conjecture holds for almost all 9-regular, 8-edge-connected graphs. These results make use of the technical small subgraph conditioning method.<\/jats:p>","DOI":"10.37236\/12513","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T19:56:34Z","timestamp":1747166194000},"source":"Crossref","is-referenced-by-count":0,"title":["Almost All 9-Regular Graphs Have a Modulo-5 Orientation"],"prefix":"10.37236","volume":"32","author":[{"given":"Michelle","family":"Delcourt","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Reaz","family":"Huq","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pawe\u0142","family":"Pra\u0142at","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,5,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T19:56:34Z","timestamp":1747166194000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,13]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/12513","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,5,13]]},"article-number":"P2.22"}}