{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:14Z","timestamp":1753893794739,"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>For any set $S$ of positive integers, a mixed hypergraph ${\\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of $\\{s,t\\}$ with $2\\leq s\\leq t-2$ is $2t-s$. Kr\u00e1l proved that there exists a one-realization of $S$ with at most $|S|+2\\max{S}-\\min{S}$ vertices. In this paper, we\u00a0 determine the number\u00a0 of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Kr\u00e1l\u00a0 in 2004.<\/jats:p>","DOI":"10.37236\/1171","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:33:46Z","timestamp":1578695626000},"source":"Crossref","is-referenced-by-count":6,"title":["The Smallest One-Realization of a Given Set"],"prefix":"10.37236","volume":"19","author":[{"given":"Ping","family":"Zhao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kefeng","family":"Diao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kaishun","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,1,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p19\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T17:57:05Z","timestamp":1579283825000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,16]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/1171","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2012,1,16]]},"article-number":"P19"}}