# Exponential sized graph 3-coloring is in MIP

I was watching a talk by Anand Natarajan on $$\text{NEEXP} \subseteq \text{MIP*}$$, and he uses $$\text{3-coloring}$$ as an example problem for $$\text{NEXP} = \text{MIP}$$ (timestamp 3:50). He mentioned that a two-prover interactive system could solve an exponential-sized graph's $$\text{3-coloring}$$ problem. I have two questions regarding this:

1. If this proves $$\text{NEXP} = \text{MIP}$$, then is this problem $$\text{NEXP-complete}$$?
2. By inputting an exponential-sized graph, will it not make the input exponential-sized by default?

1. This is meant to use a succinct encoding of the graph. That is, a graph on $$2^n$$ vertices $$V=\{0,1\}^n$$ with an edge set $$E\subseteq\{0,1\}^n\times\{0,1\}^n$$ is represented by a Boolean circuit $$C$$ in $$2n$$ variables such that $$(\vec a,\vec b)\in E\iff C(\vec a,\vec b)=1$$ for all $$\vec a,\vec b\in\{0,1\}^n$$. Ideally, you should think of $$C$$ having size polynomial in $$n$$. (Only a few graphs have a representation with such a small $$C$$, but this will be enough for the argument.)