  The foldings of a connected graph $G$ are defined as follows. First, $G$ is a
folding of itself. Let $G'$ be a graph obtained from $G$ by identifying two
vertices at distance 2 in $G$. Then every folding of $G'$ is a folding of $G$.
The folding number of $G$ is the minimum order of a complete folding of $G$.
Theorem: The folding number of every graph equals its chromatic number.
