Let $G=(V,E)$ be a finite simple, undirected graph. Given $f:V\to \mathbb{Z}$ we define the neighborhood sum function $\mathrm{nsum}_f:V\to\mathbb{Z}$ by setting
$\mathrm{nsum}_f(v) = \sum\{f(w):w\in N(v)\}$ for all $v\in V$.
We say that a graph $G=(V,E)$ is sum-balanceable if there is an injective function $f:V\to\mathbb{Z}$ such that $\mathrm{nsum}_f(v) = 0$ for all $v\in V$.
Is there for every positive integer $n\in\mathbb{N}$ a sum-balanceable graph $G$ with $\chi(G) =n$?