{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:13Z","timestamp":1753893853245,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We consider cell colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $\\Gamma(G)$ in the plane with only finitely many crossings, we define a cell $k$-coloring\u00a0of $\\Gamma(G)$ to be a coloring of the maximal connected regions of the drawing, the cells, with $k$ colors such that adjacent cells have different colors. By the $4$-color theorem, every drawing of a bridgeless graph has a cell $4$-coloring. A drawing of a graph is cell $2$-colorable if and only if the underlying graph is Eulerian. We show that every graph without degree 1 vertices admits a cell $3$-colorable drawing. This leads to the natural question which abstract\u00a0graphs have the property that each of their\u00a0drawings has a cell $3$-coloring. We say that such a graph is universally cell $3$-colorable. We show that every $4$-edge-connected graph and every graph admitting a nowhere-zero $3$-flow is universally cell\u00a0$3$-colorable. We also discuss circumstances under which universal cell $3$-colorability guarantees the existence of a nowhere-zero $3$-flow. On the negative side, we present an infinite family of universally cell $3$-colorable graphs without a nowhere-zero $3-flow. On the positive side, we formulate a conjecture which has a surprising relation to a famous open problem by Tutte known as the $3$-flow-conjecture. We prove our conjecture for subcubic and for $K_{3,3}$-minor-free graphs.<\/jats:p>","DOI":"10.37236\/9808","type":"journal-article","created":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T03:11:55Z","timestamp":1643771515000},"source":"Crossref","is-referenced-by-count":0,"title":["Coloring Drawings of Graphs"],"prefix":"10.37236","volume":"29","author":[{"given":"Christoph","family":"Hertrich","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Felix","family":"Schr\u00f6der","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Raphael","family":"Steiner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2022,1,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T03:13:02Z","timestamp":1643771582000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i1p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1,27]]}},"URL":"https:\/\/doi.org\/10.37236\/9808","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,1,28]]},"article-number":"P1.17"}}