Book thickness of optimal 1-planar graphs

Prove that every optimal 1-planar graph has book thickness four.

Background

The introduction identifies the book-thickness problem as an unresolved question concerning optimal 1-planar graphs. Bekos, Kaufmann, and Zielke formulated the conjecture that four pages suffice for a book embedding of every optimal 1-planar graph. Establishing this would determine the maximum book thickness within this graph class and confirm the cited conjecture.

References

Moreover, to the best of our knowledge, several interesting problems concerning optimal 1-planar graphs remain unresolved. For example, Bekos et al. conjectured that every optimal 1-planar graph has book thickness four .

— On the Restricted Edge-Cuts of Optimal 1-Planar Graphs  (2608.13015 - Zhang et al., 13 Aug 2026) in Section 1, Introduction