The Smallest Realization of a Given Vector

Abstract

For a vector \({R} = (r_1, r_2, \ldots, r_m)\) of non-negative integers, a mixed hypergraph \(\mathcal{H}\) is a realization of \({R}\) if its chromatic spectrum is \({R}\). In this paper, we determine the minimum number of vertices of realizations of a special kind of vectors \({R}_2\). As a result, we partially solve an open problem proposed by Král in \(2004\).