A Hajós-like theorem for weighted coloring

The Hajós' Theorem (Wiss Z Martin Luther Univ Math-Natur Reihe, 10, pp 116-117, 1961) shows a necessary and sufficient condition for the chromatic number of a given graph $G$ to be at least $k$ : $G$ must contain a $k$ -constructible subgraph. A graph is $k$ -constructible if it can be obtained from a complete graph of order $k$ by successively applying a set of well-defined operations. Given a vertex-weighted graph $G$ and a (proper) $r$ -coloring $c=\{C_1, \ldots , C_r\}$ of $G$ , the weight of a color class $C_i$ is the maximum weight of a vertex colored $i$ and the weight of $c$ is the sum of the weights of its color classes. The objective of the Weighted Coloring Problem [7] is, given a vertex-weighted graph $G$ , to determine the minimum weight of a proper coloring of $G$ , that is, its weighted chromatic number. In this article, we prove that the Weighted Coloring Problem admits a version of the Hajós' Theorem and so we show a necessary and sufficient condition for the weighted chromatic number of a vertex-weighted graph $G$ to be at least $k$ , for any positive real $k$.