Hardness of approximating acyclic chromatic number
It is not a gap-preserving reduction, but an approximation factor preserving reduction. The comment by Manuel Lafond is very close to an answer (but I cannot concur with the opinion that having same optimal values suffices; that is why I am writing this as an answer)

Quoting from Vazirani (the notation $s_1,s_2$ is mine):


  Let $\Pi_1$ and $\Pi_2$ be two minimization problems. An approximation factor preserving reduction from $\Pi_1$ to $\Pi_2$ consists of two polynomial time algorithms, $f$ and $g$, such that
  
  
  for any instance $I_1$ of $\Pi_1$, $I_2 = f(I_1)$ is an instance of $\Pi_2$ such that
  OPT$_{\Pi_2}(I_2) \leq$ OPT$_{\Pi_1}(I_1)$, and
  for any solution $s_2$ of $I_2$, $s_1= g(I_1 ,s_2)$ is a solution of $I_1$ such that
  obj$_{\Pi_1}(I_1,s_1) \leq$ obj$_{\Pi_2}(I_2,s_2)$.
  


Note: obj$_{\Pi_1}(I_1,s_1)$ simply means the value of the objective function for instance $I_1$ for solution $s_1$. Here, for example it is the number of colours used by a colouring($s_1$) of a graph($I_1$).

The whole point of this reduction is that $\chi(G)=\chi_a(H)$. For $3\leq p\leq \Delta$, it is proved that $\chi(G)\leq p\iff \chi_a(H)\leq p$. Also, $\chi(G)\leq \Delta+1$ and $\chi_a(H)\leq \Delta+1$ (I wonder why they didn't mention the latter non-trivial fact in the proof). The choice of exactly $\Delta$ multilpe edges in the construction of $H$ plays a big role here.

I am told in person by someone working in the area that OPT$_{\Pi_1}(I_1)=$ OPT$_{\Pi_2}(I_2)$ suffices. But, I think this is not the case. If this were true, the definition of approximation factor preserving reduction can be simplified to the simple condition OPT$_{\Pi_1}(I_1)=$ OPT$_{\Pi_2}(I_2)$. Note that the 2nd condition implies that OPT$_{\Pi_1}(I_1) \leq$ OPT$_{\Pi_2}(I_2)$. I guess there must be a reason why they used a stronger condition in the definition. (As an evidence, many other approximation related reductions also demand mapping between objective value of solutions.)