The smallest one-realization of a given set

Abstract

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. \cite{Jiang} showed that the minimum number of vertices of realization of $\{s,t\}$ with $2\le s\le t-2$ is $2t-s$. Kr$\acute{\mathrm{a}}$l \cite{Kral} proved that there exists a one-realization of $S$ with at most $|S|+2\max S-\min S$ vertices. In this paper, we improve Kr$\acute{\mathrm{a}}$l's result, and determine the size 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$\acute{\mathrm{a}}$l in 2004.