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 $\chi(G) =n$?



Sum-balanceable finite graphs
Consider the graph with vertices $\{x,y\}\times\{ 1,2,...n\}$, and every two vertices connected unless they are of the form $(x,a),(y,a)$ $(a\in\{1,2,...n\})$.


The graph have chromatic number $n$, since the graph is perfect and its clique number is $n$.


Noticing that the vertex $(x,a)$ is connected to $(x,k)$ and $(y,k)$ for $k\neq a$, the graph is sum-balanceable since we can assign $k$ to vertex $(x,k)$ and $-k$ to vertex $(y,k)$ for $k$ in $\{1,2,...n\}$. In fact, any $f$ satisfying $ ∀k:f((x,k))=-f((y,k))$ would work.


