The smallest one-realization of a given set

TL;DR

This paper determines the minimal size of a one-realization of a given set in mixed hypergraphs, improving previous bounds and partially solving open problems in the field.

Contribution

It establishes the exact size of the smallest one-realization for any set, advancing understanding of hypergraph realizations and resolving longstanding open questions.

Findings

Derived the minimal number of vertices for one-realizations of any set.

Improved previous upper bounds on the size of such realizations.

Partially solved open problems from Jiang et al. and Král.

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 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 $2\leq s\leq t-2$ is $2t-s$. Kr$\acute{\rm a}$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$\acute{\rm a}$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$\acute{\rm a}$l in 2004.