# Increasing the chromatic number by “folding” two vertices of distance 2

Is there a finite, connected, simple, undirected graph $G=(V,E)$ such that

1. $G$ is not complete, and
2. whenever two vertices of distance $2$ are identified ("folded"), then the chromatic number increases?

We may assume $G$ is not complete. If $G$ is a cycle, then identifying any two vertices at distance 2 does not change the chromatic number. Now assume that $G$ is not a cycle. Consider a colouring of $G$ with $k:=\chi(G)$ colours. Let $v$ be a vertex of maximum degree $d$. By Brooks' Theorem, $k \leqslant d$. At most $k-1 \leqslant d-1$ colours appear on the neighbours of $v$. So there exists distinct neighbours $x,y$ of $v$ with the same colour. Identifying $x$ and $y$ produces a $k$-colourable graph. So the chromatic number has not increased.