The chromatic spectrum of 3-uniform bi-hypergraphs

TL;DR

This paper constructs 3-uniform bi-hypergraphs with prescribed chromatic spectra, solving an open problem and providing bounds on the minimum vertices needed for given feasible sets.

Contribution

It introduces a method to realize any feasible set as the chromatic spectrum of 3-uniform bi-hypergraphs, addressing an open problem from 2008.

Findings

Constructed hypergraphs with arbitrary feasible sets and spectra

Solved an open problem on chromatic spectra of 3-uniform bi-hypergraphs

Established bounds on the minimum number of vertices for given feasible sets

Abstract

Let $S=\{n_1,n_2,...,n_t\}$ be a finite set of positive integers with $\min(S)\geq 3$ and $t\geq 2$. For any positive integers $s_1,s_2,...,s_t$, we construct a family of 3-uniform bi-hypergraphs ${\cal H}$ with the feasible set $S$ and $r_{n_i}=s_i, i=1,2,...,t$, where each $r_{n_i}$ is the $n_i$th component of the chromatic spectrum of ${\cal H}$. As a result, we solve one open problem for 3-uniform bi-hypergraphs proposed by Bujt\'{a}s and Tuza in 2008. Moreover, we find a family of sub-hypergraphs with the same feasible set and the same chromatic spectrum as it's own. In particular, we obtain a small upper bound on the minimum number of vertices in 3-uniform bi-hypergraphs with any given feasible set.