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Oporowski, Bogdan


Graph Theory

Book Thickness of Subdivisions ★★

Author(s): Blankenship; Oporowski

Let $ G $ be a finite undirected simple graph.

A $ 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 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.

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

Keywords: book embedding; book thickness

Posted by David Wood
updated July 10th, 2021
1 comment
Graph Theory » Topological G.T. » Coloring

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

Author(s): Oporowski; Zhao

For a graph $ G $, we let $ cr(G) $ denote the crossing number of $ G $, and we let $ \omega(G) $ denote the size of the largest complete subgraph of $ G $.

Question   Does every graph $ G $ with $ cr(G) \le 5 $ and $ \omega(G) \le 5 $ have a 5-coloring?

Keywords: coloring; crossing number; planar graph

Posted by mdevos
updated October 9th, 2007
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Graph Theory » Topological G.T. » Coloring

Bounded colorings for planar graphs ★★

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

Question   Does there exists a fixed function $ f : {\mathbb N} \rightarrow {\mathbb N} $ so that every planar graph of maximum degree $ d $ has a partition of its vertex set into at most three sets $ \{V_1,V_2,V_3\} $ so that for $ i=1,2,3 $, every component of the graph induced by $ V_i $ has size at most $ f(d) $?

Keywords: coloring; partition; planar graph

Posted by mdevos
updated May 23rd, 2007
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