{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T03:47:58Z","timestamp":1761709678028,"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>A  mixed hypergraph is a triple ${\\cal H} = (V,{\\cal C}, {\\cal D})\\;$ where $V$ is the vertex set and  ${\\cal C}$ and ${\\cal D}$ are families of subsets of $V$, the ${\\cal C}$-edges and ${\\cal D}$-edges, respectively.  A $k$-colouring of ${\\cal H}$ is a mapping $c: V\\rightarrow [k]$ such that each ${\\cal C}$-edge has at least two vertices with a ${\\cal C}$ommon colour and each ${\\cal D}$-edge has at least two vertices of ${\\cal D}$ifferent colours. ${\\cal H}$ is called a planar mixed hypergraph if its bipartite representation is a planar graph. Classic graphs are the special case of mixed hypergraphs when ${\\cal C}=\\emptyset$ and all the ${\\cal D}$-edges have size 2, whereas in a bi-hypergraph ${\\cal C} = {\\cal D}$. We investigate the colouring properties of planar mixed hypergraphs. Specifically, we show that maximal planar bi-hypergraphs are 2-colourable, find formulas for their chromatic polynomial and chromatic spectrum in terms of 2-factors in the dual, prove that their chromatic spectrum is gap-free and provide a sharp estimate on the maximum number of colours in a colouring.<\/jats:p>","DOI":"10.37236\/1538","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:02:42Z","timestamp":1578708162000},"source":"Crossref","is-referenced-by-count":18,"title":["Colouring Planar Mixed Hypergraphs"],"prefix":"10.37236","volume":"7","author":[{"given":"Andr\u00e9","family":"K\u00fcndgen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eric","family":"Mendelsohn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vitaly","family":"Voloshin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2000,9,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r60\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r60\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:21:47Z","timestamp":1579324907000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v7i1r60"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,9,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2000,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1538","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2000,9,28]]},"article-number":"R60"}}