Is the acyclic chromatic number bounded in terms of the book thickness? 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?
 A: 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.

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
$\overline{\theta}(G)$.
A: Here's a simple and self-contained answer: 
As you have said, the chromatic number is bounded by book thickness, say, book thickness $≤n \implies$ chromatic number $≤f(n)$.
For a graph with book thickness $n$, color it with $f(n)$ colors, and embed it on $n$ pages. Each page contains an outerplanar graph by definition, and outerplanar graphs has bounded acyclic chromatic number, so the whole graph can be acyclically colored with $3f(n)$ colors.
So  book thickness $≤n \implies$ acyclic chromatic number $≤3f(n)$.
