Let $G=(V,E)$ be a simple graph. Let $w$ be a non-negative, integer valued weight function on the vertex set. The chromatic number $\chi(G,w)$ of the vertex-weighted graph $(G,w)$ is defined to be the minimum number of colors needed so that each vertex $v$ is assigned $w(v)$ colors and the color sets assigned to adjacent vertices are disjoint. By generalizing the well-known greedy coloring algorithm to weighted graphs, one obtains the upper bound $\chi(G,w) \le \max_{v \in V} \{w(v)+w(N(v))\}$, where $w(N(v))$ denotes the sum of the weights of the vertices that are neighbors of $v$.

What is a similar upper bound for the weighted chromatic number of a hypergraph? Any results or pointers to the literature would be helpful.

The definition of the chromatic number of a weighted hypergraph is as follows. Let $H=(V, \mathcal{E})$ be a hypergraph with ground set $V$ and a set $\mathcal{E}$ of hyperedges. A proper coloring of the hypergraph is an assignment of a color to each vertex such that no hyperedge is monochromatic. This is generalized to weighted hypergraphs as follows. Let $w$ be a non-negative, integer valued weight function on $V$. The chromatic number $\chi(H,w)$ is the minimum number of colors needed so that each vertex $v$ is assigned $w(v)$ colors and no color class contains a hyperedge.