Let $\mathbb{N}$ denote the set of positive integers, and 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 $v\in V$.

My goal is to prove that sum-balanceability is computable. One way to do this is to show that in order to find an injective $f:V\to\mathbb{Z}$ with the desired property, we only need to check finitely many combinations for assignments of $v\in V$ to an integer. Let's make this hand-waving formal.

**Question.** Is there a computable function $b:\mathbb{N}\to\mathbb{N}$ with the following property?

Whenever $G=(V,E)$ with $|V| = n\in\mathbb{N}$ is sum-balanceable, there is an injective function $f:V\to \mathbb{Z}$ with $\mathrm{nsum}_f(v) = 0$ for all $v\in V$ and $$|f(v)| \leq b(n)$$ for all $v\in V$.

Note that the existence of such a computable function $b:\mathbb{N}\to\mathbb{N}$ would directly imply that sum-balanceability is computable.