{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:38Z","timestamp":1753893818228,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A graph $G$ is uniquely $k$-colorable\u00a0if the chromatic number of $G$ is $k$\u00a0and $G$ has only one $k$-coloring up to permutation of the colors.\u00a0A uniquely $k$-colorable graph $G$ is edge-critical\u00a0if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\\in E(G)$.\u00a0In this paper,\u00a0we prove that if $G$ is\u00a0an edge-critical uniquely $3$-colorable planar graph,\u00a0then $|E(G)|\\leq \\frac{8}{3}|V(G)|-\\frac{17}{3}$.\u00a0On the other hand,\u00a0there exists\u00a0an infinite family of edge-critical uniquely 3-colorable planar graphs with $n$ vertices and $\\frac{9}{4}n-6$ edges.\u00a0Our result gives a first non-trivial upper bound for $|E(G)|$.<\/jats:p>","DOI":"10.37236\/3228","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:21:52Z","timestamp":1578705712000},"source":"Crossref","is-referenced-by-count":4,"title":["The Size of Edge-critical Uniquely 3-Colorable Planar Graphs"],"prefix":"10.37236","volume":"20","author":[{"given":"Naoki","family":"Matsumoto","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,9,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p49\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:12:45Z","timestamp":1579259565000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i3p49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,26]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2013,7,19]]}},"URL":"https:\/\/doi.org\/10.37236\/3228","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,9,26]]},"article-number":"P49"}}