
Ping Zhao

Kefeng Diao

Kaishun Wang
Abstract
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 onerealization 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 $2\leq s\leq t2$ is $2ts$. Král proved that there exists a onerealization of $S$ with at most $S+2\max{S}\min{S}$ vertices. In this paper, we determine the number of vertices of the smallest onerealization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Král in 2004.