The Smallest Uniform Color-Bounded Hypergraphs Which are One-Realizations of a Given Set

Abstract

A color-bounded hypergraph is a hypergraph with vertex set X and edge set ${\mathcal {E}}=\{E_1,E_2,\dots ,E_m\}$, together with integers $s_i$ and $t_i$ ($1\le s_i\le t_i\le |E_i|$) for $i=1,2,\ldots ,m$. A vertex coloring $\varphi$ is proper if the number of colors occurring in edge $E_i$ satisfies $s_i\le |\varphi (E_i)|\le t_i$, for every $1\le i\le m$. If $s_i=s$ and $t_i=t$ for all i, we simply denote the color-bounded hypergraph by ${\mathcal {H}}=(X, {\mathcal {E}},s,t)$. A set of positive integers $\Phi (\mathcal {H})$ is called feasible, if it consists of all k for which there exists a proper coloring of $\mathcal {H}$ using precisely k colors. Chromatic spectrum of a hypergraph $\mathcal {H}$ is a vector with each entry $r_k$ equal to the number of partitions of vertex set induced by all proper colorings using k colors. Let S be a finite set of positive integers. A color-bounded hypergraph 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 this paper, we determine the minimum number of vertices of r-uniform color-bounded hypergraphs ${\mathcal {H}}=(X, {\mathcal {E}},2,t)$ which are one-realizations of S for the case when $\lceil \frac{r}{2}\rceil <t\le r-2$ and $\max (S)\ge \frac{3r}{2}$.