Book Thickness of Subdivisions

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A $k$-page book embedding of a finite undirected simple graph $G$ is a linear order $\prec$ of its vertices and an assignment of edges to $k$ pages such that no two same-page edges $vw,xy$ have endpoints ordered $v\prec x\prec w\prec y$. The book thickness $\operatorname{bt}(G)$ is the least nonnegative number of pages required. Let $G'$ replace each edge $uv$ by $u w_{uv},w_{uv}v$, using a distinct new vertex for each edge.

Statement recorded in the supplied source.

Is there a function $f:\{0,1,2,\ldots\}\to\{0,1,2,\ldots\}$ such that \[\operatorname{bt}(G)\leq f\left(\operatorname{bt}(G')\right)\] for every finite undirected simple graph $G$? [S2]

Short English statement

Can the book thickness of a graph be bounded solely in terms of the book thickness after every edge is subdivided exactly once?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3105 (OPG-34908). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Blankenship, Robin; Oporowski, Bogdan, Book Thickness of Subdivisions, Open Problem Garden, node 34908. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/book_thickness_of_subdivisions.