Graph Theory » Topological G.T. » Coloring

3-Colourability of Arrangements of Great Circles ★★

Author(s): Felsner; Hurtado; Noy; Streinu

Consider a set $ S $ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $ S $ has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.

Conjecture   Every arrangement graph of a set of great circles is $ 3 $-colourable.

Keywords: arrangement graph; graph coloring

Posted by David Wood
updated October 22nd, 2020
2 comments
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 » Coloring » Vertex coloring

Double-critical graph conjecture ★★

Author(s): Erdos; Lovasz

A connected simple graph $ G $ is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.

Conjecture   $ K_n $ is the only $ n $-chromatic double-critical graph

Keywords: coloring; complete graph

Posted by DFR
updated November 30th, 2009
2 comments