{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:08:02Z","timestamp":1758823682683,"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>Assume $G$ is a graph and $S$ is a\u00a0set\u00a0of permutations of positive\u00a0integers. An $S$-signature of $G$ is a pair $(D, \\sigma)$, where $D$ is an orientation of $G$ and $\\sigma: E(D) \\to S$ is a mapping which assigns to each arc $e=(u,v)$ a permutation $\\sigma(e)$ in $S$. We say $G$ is $S$-$k$-colourable if for any $S$-signature $(D, \\sigma)$ of $G$, there is a mapping $f: V(G) \\to [k]$ such that for each arc $e=(u,v)$ of $G$, $\\sigma(e)(f(u)) \\ne f(v)$. The concept of $S$-$k$-colourable is a common generalization of many colouring concepts. This paper studies the problem as to which subsets $S$ of $S_4$, every\u00a0 planar graph is $S$-$4$-colourable. We call\u00a0such\u00a0a subset $S$ of $S_4$ a good subset. The Four Colour Theorem is equivalent to saying that $S=\\{id\\}$ is good. It was proved by Jin, Wong and Zhu (arXiv:1811.08584) that a subset $S$ containing $id$ is good if and only if $S=\\{id\\}$. In this paper, we prove that, up to conjugation, every good subset of $S_4$ not containing $id$ is a subset of $\\{(12),(34),(12)(34)\\}$.<\/jats:p>","DOI":"10.37236\/9338","type":"journal-article","created":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:34Z","timestamp":1599187654000},"source":"Crossref","is-referenced-by-count":2,"title":["$4$-Colouring of Generalized Signed Planar Graphs"],"prefix":"10.37236","volume":"27","author":[{"given":"Yiting","family":"Jiang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuding","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,8,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p31\/8150","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p31\/8150","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:34Z","timestamp":1599187654000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i3p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,21]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,7,9]]}},"URL":"https:\/\/doi.org\/10.37236\/9338","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,8,21]]},"article-number":"P3.31"}}