Book Thickness of Subdivisions
Let be a finite undirected simple graph.
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and 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 book thickness of , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
The conjecture is due to [B099]. The conjecture is true for complete graphs [BO99,EM99,E02]. The conjecture is discussed in depth in [DW05].
*[BO99] Robin Blankenship and Bogdan Oporowski. 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. Stacks, queues and tracks: Layouts of graph subdivisions. Discrete Mathematics & Theoretical Computer Science 7:155-202, 2005.
[EM99] Hikoe Enomoto and Miki Shimabara Miyauchi. Embedding graphs into a three page book with crossings of edges over the spine. SIAM J. Discrete Math., 12(3):337–341, 1999.
[E02] David Eppstein. 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.