**Question.** If $G$ is a Hamiltonian, does it contain a chromatic path visiting all the vertices? (I define the term "chromatic path" below.)

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We denote by $\mathbb{N}$ the set of positive integers and set $[n] = \{1,\ldots,n\}$ for $n\in\mathbb{N}$.

Let $G= (V,E)$ be a simple undirected graph on $n>1$ vertices, and let $b:[n]\to V$ be a bijection. We assign to it the map $c_b:[n] \to [n]$, which is recursively defined by:

- $c_b(1) = 1$;
- if $k\in[n]$ and $k>1$ let 
$$c_b(k) = \min\:\big(\mathbb{N}\setminus\{c_b(j): j \in [k-1]\land \{b(j),b(k)\}\in E\}\big).$$

We call $b$ *chromatic* if $\text{im}(b) = [\chi(G)]$. For every graph there is a chromatic bijection (see [here][1]). A *chromatic path* is a chromatic bijection that is also a [path][2].


  [1]: https://en.wikipedia.org/wiki/Greedy_coloring#Optimal_ordering
  [2]: https://en.wikipedia.org/wiki/Path_(graph_theory)