Book Thickness of Subdivisions

Importance: Medium ✭✭
Subject: Graph Theory
Recomm. for undergrads: yes
Posted by: David Wood
on: January 19th, 2009

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}

The conjecture is due to [B099]. The conjecture is true for complete graphs [BO99,EM99,E02]. The conjecture is discussed in depth in [DW05].

Bibliography

*[BO99] Robin Blankenship and Bogdan Oporowski. \emph{Drawing Subdivisions Of Complete And Complete Bipartite Graphs On Books}, Technical Report 1999-4, Department of Mathematics, Louisiana State University, 1999.

[DW05] Vida Dujmovic and David Wood. \href[\emph{Stacks, queues and tracks: Layouts of graph subdivisions}]{http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/67}. Discrete Mathematics & Theoretical Computer Science 7:155-202, 2005.

[EM99] Hikoe Enomoto and Miki Shimabara Miyauchi. \href[\emph{Embedding graphs into a three page book with $O(M \log N)$ crossings of edges over the spine}]{http://dx.doi.org/10.1137/S0895480195280319}. SIAM J. Discrete Math., 12(3):337–341, 1999.

[E02] David Eppstein. \emph{Separating thickness from geometric thickness}. In Proc. 10th International Symp. on Graph Drawing (GD ’02), pp. 150–161. vol. 2528 of Lecture Notes in Comput. Sci. Springer, 2002.


* indicates original appearance(s) of problem.