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 \(\max (S)\ge \frac{3r}{2}\).