# Oporowski, Bogdan

## Book Thickness of Subdivisions ★★

Author(s): Blankenship; Oporowski

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.

Conjecture   There is a function such that for every graph ,

Keywords: book embedding; book thickness

## 5-coloring graphs with small crossing & clique numbers ★★

Author(s): Oporowski; Zhao

For a graph , we let denote the crossing number of , and we let denote the size of the largest complete subgraph of .

Question   Does every graph with and have a 5-coloring?

Keywords: coloring; crossing number; planar graph

## Bounded colorings for planar graphs ★★

Author(s): Alon; Ding; Oporowski; Vertigan

Question   Does there exists a fixed function so that every planar graph of maximum degree has a partition of its vertex set into at most three sets so that for , every component of the graph induced by has size at most ?

Keywords: coloring; partition; planar graph