book embedding

Book Thickness of Subdivisions ★★

Author(s): Blankenship; Oporowski

Let $G$ be a finite undirected simple graph.

A \emph{$k$-page book embedding} of $G$ consists of a linear order $\preceq$ of $V(G)$ and a (non-proper) $k$-colouring of $E(G)$ such that edges with the same colour do not cross with respect to $\preceq$. That is, if $v\prec x\prec w\prec y$ for some edges $vw,xy\in E(G)$, then $vw$ and $xy$ receive distinct colours.

One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.

The \emph{book thickness} of $G$, denoted by bt$(G)$ is the minimum integer $k$ for which there is a $k$-page book embedding of $G$.

Let $G'$ be the graph obtained by subdividing each edge of $G$ exactly once.

\begin{conjecture} There is a function $f$ such that for every graph $G$, $$\text{bt}(G) \leq f( \text{bt}(G') )\enspace.$$ \end{conjecture}

Keywords: book embedding; book thickness

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