The smallest one-realization of a given set IV

TL;DR

This paper generalizes previous results on the minimal size of 3-uniform bi-hypergraphs that realize a set of integers, extending the findings to r-uniform bi-hypergraphs, and characterizes their minimal vertex counts.

Contribution

It extends the characterization of the smallest one-realization hypergraphs from 3-uniform to r-uniform bi-hypergraphs, broadening the scope of previous work.

Findings

Determined the minimum number of vertices for r-uniform bi-hypergraphs as one-realizations.

Generalized previous 3-uniform results to r-uniform cases.

Provides formulas or bounds for minimal vertex counts in r-uniform hypergraphs.

Abstract

Let $S$ be a finite set of positive integers. A mixed hypergraph ${\cal H}$ is a one-realization of $S$ if its feasible set is $S$ and each entry of its chromatic spectrum is either 0 or 1. In [P. Zhao, K. Diao, Y. Chang and K. Wang, The smallest one-realization of a given set \uppercase\expandafter{\romannumeral2}, Discrete Math. 312 (2012) 2946--2951], we determined the minimum number of vertices of a 3-uniform bi-hypergraph which is a one-realization of $S$. In this paper, we generalize this result to $r$-uniform bi-hypergraphs.