Triangle-free intersection graphs of line segments with large chromatic number

In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer $k$, we construct a triangle-free family of line segments in the plane with chromatic number greater than $k$. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.Comment: Small corrections, bibliography updat

Coloring intersection graphs of x-monotone curves in the plane

A class of graphs G is χ-bounded if the chromatic number of the graphs in G is bounded by some function of their clique number. We show that the class of intersection graphs of simple families of x-monotone curves in the plane intersecting a vertical line is χ-bounded. As a corollary, we show that the class of intersection graphs of rays in the plane is χ-bounded, and the class of intersection graphs of unit segments in the plane is χ-bounded.National Science Foundation (U.S.) (Postdoctoral Fellowship

Coloring curves that cross a fixed curve

We prove that for every integer $t\geq 1$, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most $t$ points is $\chi$-bounded. This is essentially the strongest $\chi$-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers $k\geq 2$ and $t\geq 1$, every $k$-quasi-planar topological graph on $n$ vertices with any two edges crossing at most $t$ times has $O(n\log n)$ edges.Comment: Small corrections, improved presentatio

Coloring curves that cross a fixed curve

We prove that for every integer t\geqslant 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is $χ$-bounded. This is essentially the strongest $χ$-boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers $k\geqslant$ 2 and $t\geqslant$ 1, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has $O(nlogn)$ edges

A survey of χ\chi-boundedness

If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now

Colouring Arcwise Connected Sets In The Plane II

. Let G be the family of finite collections S where S is a collection of closed, bounded, arcwise connected sets in R 2 which for any S; T 2 S where S " T 6= ;, it holds that S " T is arcwise connected. Given S 2 G which is triangle-free, we show that provided S is sufficiently large there exists a subcollection S 0 ae S of at most 5 sets with the property that the sets of S surrounded by S 0 induce an intersection graph H where Ø(H) 1 ff Ø(G), and ff ? 1 does not depend on S. In conjuction with this result, we obtain a new result concerning the socalled L-graph conjecture. We show that if for a triangle-free collection of L-shapes L it holds that for any two intersecting L-shapes the ratio of their horizontal lengths and the ratio of their vertical lengths are bounded above by a constant fl independent of L, then the chromatic number of the L-graph G(L) is bounded above by a constant depending only on fl. 0. Introduction Any of the notation and concepts not explicitly def..