2,625 research outputs found
On colouring point visibility graphs
In this paper we show that it can be decided in polynomial time whether or
not the visibility graph of a given point set is 4-colourable, and such a
4-colouring, if it exists, can also be constructed in polynomial time. We show
that the problem of deciding whether the visibility graph of a point set is
5-colourable, is NP-complete. We give an example of a point visibility graph
that has chromatic number 6 while its clique number is only 4
On Visibility and Blockers
This expository paper discusses some conjectures related to visibility and
blockers for sets of points in the plane
Colouring quadrangulations of projective spaces
A graph embedded in a surface with all faces of size 4 is known as a
quadrangulation. We extend the definition of quadrangulation to higher
dimensions, and prove that any graph G which embeds as a quadrangulation in the
real projective space P^n has chromatic number n+2 or higher, unless G is
bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996),
219-227]. The family of quadrangulations of projective spaces includes all
complete graphs, all Mycielski graphs, and certain graphs homomorphic to
Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser
theorem
On Colourability of Polygon Visibility Graphs
We study the problem of colouring the visibility graphs of polygons. In particular, we provide
a polynomial algorithm for 4-colouring of the polygon visibility graphs, and prove that the 6-
colourability question is already NP-complete for them
Colouring Polygon Visibility Graphs and Their Generalizations
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time
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