{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:17Z","timestamp":1753893797625,"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 $H$ is a triple $(V,{\\cal C},{\\cal D})$ where $V$ is the vertex set and ${\\cal C}$ and ${\\cal D}$ are families of subsets of $V$, called ${\\cal C}$-edges and ${\\cal D}$-edges. A vertex coloring of $H$ is proper if each ${\\cal C}$-edge contains two vertices with the same color and each ${\\cal D}$-edge contains two vertices with different colors.  The spectrum of $H$ is a vector $(r_1,\\ldots,r_m)$ such that there exist exactly $r_i$ different colorings using exactly $i$ colors, $r_m\\ge 1$ and there is no coloring using more than $m$ colors.  The feasible set of $H$ is the set of all $i$'s such that $r_i\\ne 0$. We construct a mixed hypergraph with $O(\\sum_i\\log r_i)$ vertices whose spectrum is equal to $(r_1,\\ldots,r_m)$ for each vector of non-negative integers with $r_1=0$.  We further prove that for any fixed finite sets of positive integers $A_1\\subset A_2$ ($1\\notin A_2$), it is NP-hard to decide whether the feasible set of a given mixed hypergraph is equal to $A_2$ even if it is promised that it is either $A_1$ or $A_2$.  This fact has several interesting corollaries, e.g., that deciding whether a feasible set of a mixed hypergraph is gap-free is both NP-hard and coNP-hard.<\/jats:p>","DOI":"10.37236\/1772","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:23:09Z","timestamp":1578720189000},"source":"Crossref","is-referenced-by-count":19,"title":["On Feasible Sets of Mixed Hypergraphs"],"prefix":"10.37236","volume":"11","author":[{"given":"Daniel","family":"Kr\u00e1l","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,3,4]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r19\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:04:30Z","timestamp":1579323870000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,3,4]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1772","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,3,4]]},"article-number":"R19"}}