The smallest one-realization of a given set (Q426777)

Summary: For any set \(S\) of positive integers, a mixed hypergraph \({\mathcal H}\) is a realization of \(S\) if its feasible set is \(S\), furthermore, \({\mathcal H}\) 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 \(2t-s\). Král proved that there exists a one-realization of \(S\) with at most \(|S|+2\max{S}-\min{S}\) vertices. In this paper, we determine the number of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by \textit{T. Jiang} et al. [Graphs Comb. 18, No. 2, 309--318 (2002; Zbl 0994.05063)] and by \textit{D. Král} [Electron. J. Comb. 11, No. 1, Research paper R19, 14 p. (2004); printed version J. Comb. 11, No. 1 (2004; Zbl 1055.05058 )].