This is a subquestion for an older question about a certain kind of greedy coloring.

Let $G = (V,E)$ be a finite, connected, simple, undirected graph. By a *roundtrip of $G$* we mean a map $r:\{0,\ldots,n\} \to V$ for some $n\in\mathbb{N}$ with the following properties:

- $r$ is surjective,
- $\{r(k), r(k+1)\} \in E$ for all $k \in \{0, \ldots, n-1\}$, and
- $r(0) = r(n)$.

An easy inductive argument shows that we can select $n$ such that $n \leq 2|G|$.

Given a roundtrip $r$ and a vertex $v\in V$, we assign a *roundtrip coloring $c_{r,v}:V\to\mathbb{N}$ of $G$, starting at $v$* in the following manner: Start at $v$, proceed along the roundtrip $r$ and greedy-color every uncolored vertex along the way, until every vertex has been colored. Let $\chi(r,v)$ be the number of colors used.

**Question.** What is an example of a connected graph $G=(V,E)$ and $v\neq v'\in V$ such that $\chi(r,v)\neq \chi(r,v')$?