If $G=(V,E)$ is a graph, by $\chi(G)$ we define the smallest cardinal $\kappa$ such that there is a coloring map $c:V\to \kappa$. 

For any coloring $c:V(G) \to \chi(G)$ we define the *set of fibers* of $c$ by $\newcommand{\Fib}{\text{Fib}}\Fib(c) = \big\{c^{-1}(\{\alpha\}):\alpha \in \kappa\big\}$. 

Note that if $V(G)$ is non-empty, then so is every member of 
$\Fib(c)$. 

We say that a coloring $c_0:V(G) \to\chi(G)$ is *maximally efficient* if $|\Fib(c_0)\setminus \Fib(c)| \leq |\Fib(c)\setminus \Fib(c_0)|$ for every coloring $c:V(G)\to \chi(G)$.

**Question.** Does every graph, finite or infinite, have a maximally efficient coloring?