Attila Pór

On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane

AbstractThe visibility graph V(P) of a point set P \subseteq R2 has vertex set P, such that two points v,w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P). We characterise the 2- and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main result is a super-polynomial lower bound on the chromatic number (in terms of the clique number).

A Step toward the Bermond-Thomassen Conjecture about Disjoint Cycles in Digraphs

In 1981, Bermond and Thomassen conjectured that every digraph with minimum out-degree at least

On the computational complexity of partial covers of Theta graphs

2k-1

On the Connectivity of Visibility Graphs

contains

Kneser Representations of Graphs

k

Colourings of the cartesian product of graphs and multiplicative Sidon sets

disjoint cycles. This conjecture is trivial for

Blocking Colored Point Sets

k=1

Paths with No Small Angles

, and was established for

Colourful and Fractional (p,q)-theorems

k=2

Destroying cycles in digraphs

by Thomassen in 1983. We verify it for the next case, proving that every digraph with minimum out-degree at least five contains three disjoint cycles. To show this, we improve Thomassens result by proving that every digraph whose vertices have out-degree at least three, except at most two with out-degree two, indeed contains two disjoint cycles.