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

Triangle-free geometric intersection graphs with no large independent sets

It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.Comment: Change of the title, minor revisio

Note on the number of edges in families with linear union-complexity

We give a simple argument showing that the number of edges in the intersection graph $G$ of a family of $n$ sets in the plane with a linear union-complexity is $O(\omega(G)n)$. In particular, we prove $\chi(G)\leq \text{col}(G)< 19\omega(G)$ for intersection graph $G$ of a family of pseudo-discs, which improves a previous bound.Comment: background and related work is now more complete; presentation improve

Burling graphs, chromatic number, and orthogonal tree-decompositions

A classic result of Asplund and Gr\"unbaum states that intersection graphs of axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmovi\'c, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note we provide a negative answer: There are graphs with arbitrarily large chromatic number for which one can find two tree-decompositions such that each bag of the first decomposition intersects each bag of the second in at most two vertices. Furthermore, this remains true even if one of the two decompositions is restricted to be a path-decomposition. This is shown using a construction of triangle-free graphs with unbounded chromatic number due to Burling, which we believe should be more widely known.Comment: v3: minor changes made following comments by the referees, v2: minor edit

Applications of a new separator theorem for string graphs

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds for string graphs.Comment: 7 page

Decomposition of multiple packings with subquadratic union complexity

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function $f(n)=o(n^2)$ with the property that any $n$ members of $\mathcal{X}$ determine at most $f(n)$ holes, which means that the complement of their union has at most $f(n)$ bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that $\mathcal{X}$ can be decomposed into at most $p$ ($1$-fold) packings, where $p$ is a constant depending only on $k$ and $f$.Comment: Small generalization of the main result, improvements in the proofs, minor correction

Triangle-free geometric intersection graphs with large chromatic number

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect and $\chi(\mathcal{F})>k$. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries, and equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.Comment: Small corrections, bibliography updat