The Smallest Realization of a Given Vector

Abstract

For a vector $R=(r_1,r_2,\dots ,r_m)$ 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 $2004$.