ISGCI says that the chromatic number of a graph is upper bounded in terms of the book thickness.
<https://www.graphclasses.org/classes/par_32.html>


This can be improved by saying that the book thickness bounds the degeneracy.


A further improvement would be that the book thickness bounds the acyclic chromatic number.


Is it true? Is there a reference?



Is the acyclic chromatic number bounded in terms of the book thickness?
I believe that "book thickness bounds the acyclic chromatic number" was
established in this paper:



> 
> Dujmovic, Vida, Attila Pór, and David R. Wood. "Track layouts of graphs." *Discrete Mathematics and Theoretical Computer Science* 6, no. 2 (2004).
>  [arXiv abs](https://arxiv.org/abs/cs/0407033).
> 
> 
> 


In the Abstract they say,



> 
> "As corollaries we
>  prove that acyclic chromatic number is bounded by both queue-number and stack-number."
> 
> 
> 


And then later (Section 5), they say, 



> 
> "Note that stack-number is also called *page-number* and *book-thickness*."
> 
> 
> 


Not sure if this is relevant to your quesiton, but
the acyclic chromatic number is not bounded by *geometric thickness*
$\overline{\theta}(G)$.


