{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:10Z","timestamp":1753893850104,"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>Representations of planar triangulations as contact graphs of a set\u00a0of internally disjoint homothetic triangles or of a set\u00a0of internally disjoint homothetic squares have received quite some\u00a0attention in recent years. In this paper we investigate\u00a0representations of planar triangulations as contact graphs of a set\u00a0of internally disjoint homothetic pentagons. Surprisingly such a\u00a0representation exists for every triangulation whose outer face is a\u00a0$5$-gon. We relate these representations to five color\u00a0forests. These combinatorial structures resemble Schnyder woods\u00a0and transversal structures, respectively. In particular there is a\u00a0bijection to certain $\\alpha$-orientations and consequently a\u00a0lattice structure on the set of five color forests of a given graph.\u00a0This lattice structure plays a role in an algorithm that is supposed\u00a0to compute a contact representation with pentagons for a given\u00a0graph. Based on a five color forest the algorithm builds a system\u00a0of linear equations and solves it, if the solution is non-negative,\u00a0it encodes distances between corners of a pentagon\u00a0representation. In this case the representation is constructed and\u00a0the algorithm terminates. Otherwise negative variables guide a\u00a0change of the five color forest and the procedure is restarted with\u00a0the new five color forest. Similar algorithms have been proposed for\u00a0contact representations with homothetic triangles and with squares.<\/jats:p>","DOI":"10.37236\/7216","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:10:28Z","timestamp":1578669028000},"source":"Crossref","is-referenced-by-count":3,"title":["Pentagon Contact Representations"],"prefix":"10.37236","volume":"25","author":[{"given":"Stefan","family":"Felsner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hendrik","family":"Schrezenmaier","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":[[2018,9,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i3p39\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i3p39\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:26:32Z","timestamp":1579235192000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i3p39"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,7]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,7,12]]}},"URL":"https:\/\/doi.org\/10.37236\/7216","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,9,7]]},"article-number":"P3.39"}}