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A grounded L-graph is the intersection graph of a collection of "L" shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number $\omega$ has chromatic number at most $17\omega^4$. This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially $\chi$-bounded. We also survey $\chi$-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.

It is proved that triangle-free intersection graphs of $n$ L-shapes in the plane have chromatic number $O(\log\log n)$. This improves the previous bound of $O(\log n)$ (McGuinness, 1996) and matches the known lower bound construction (Pawlik et al., 2013).

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space $P^n$ is at least $(n+2)$-chromatic, unless it is bipartite. They conjectured that for any integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $SG(n,k)$ contains a spanning subgraph which is a quadrangulation of $P^{n-2k}$. The purpose of this paper is to prove the conjecture.

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 presentation

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 revision

An outerstring graph is an intersection graph of curves that lie in a common half-plane and have one endpoint on the boundary of that half-plane. We prove that the class of outerstring graphs is $\chi$-bounded, which means that their chromatic number is bounded by a function of their clique number. This generalizes a series of previous results on $\chi$-boundedness of outerstring graphs with various additional restrictions on the shape of curves or the number of times the pairs of curves can cross. The assumption that each curve has an endpoint on the boundary of the half-plane is justified by the known fact that triangle-free intersection graphs of straight-line segments can have arbitrarily large chromatic number., Comment: Introduction extended by a survey of results on (outer)string graphs, some minor corrections

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number., Comment: Minor changes + some additional references not included in the journal version

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number $O(\log\log n)$, improving on the previous bound of $O(\log n)$. To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with $O(\log\log n)$ colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition)., Comment: Minor revision

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 update

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 update