  In a dispersable book embedding, the vertices of a given graph $G$ must be
ordered along a line l, called spine, and the edges of G must be drawn at
different half-planes bounded by l, called pages of the book, such that: (i) no
two edges of the same page cross, and (ii) the graphs induced by the edges of
each page are 1-regular. The minimum number of pages needed by any dispersable
book embedding of $G$ is referred to as the dispersable book thickness $dbt(G)$
of $G$. Graph $G$ is called dispersable if $dbt(G) = \Delta(G)$ holds (note
that $\Delta(G) \leq dbt(G)$ always holds).
  Back in 1979, Bernhart and Kainen conjectured that any $k$-regular bipartite
graph $G$ is dispersable, i.e., $dbt(G)=k$. In this paper, we disprove this
conjecture for the cases $k=3$ (with a computer-aided proof), and $k=4$ (with a
purely combinatorial proof). In particular, we show that the Gray graph, which
is 3-regular and bipartite, has dispersable book thickness four, while the
Folkman graph, which is 4-regular and bipartite, has dispersable book thickness
five. On the positive side, we prove that 3-connected 3-regular bipartite
planar graphs are dispersable, and conjecture that this property holds, even if
3-connectivity is relaxed.
