Coloring intersection graphs of x-monotone curves in the plane

A class of graphs G is chi-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 x-monotone curves in the plane intersecting a vertical line is chi-bounded. As a corollary we show that the class of intersection graphs of rays in the plane is chi-bounded, and the class of intersection graphs of unit segments in the plane is chi-bounded

💡 Research Summary

The paper investigates the chromatic properties of a geometric class of intersection graphs and establishes that this class is χ‑bounded, meaning that the chromatic number χ(G) of any graph G in the class can be bounded by a function of its clique number ω(G). The objects under consideration are simple x‑monotone curves in the Euclidean plane that all intersect a common vertical line L. A curve is called x‑monotone if its projection onto the x‑axis is injective (the curve never turns back in the x‑direction), and it is simple if it does not intersect itself. For each such curve we create a vertex, and we connect two vertices by an edge whenever the corresponding curves intersect. The resulting graph is called the intersection graph of the family of curves.

The main theorem states that the family of all such intersection graphs is χ‑bounded. In other words, there exists a universal function f such that for every graph G arising from a family of simple x‑monotone curves intersecting a common vertical line, we have χ(G) ≤ f(ω(G)). The proof does not give a tight closed‑form expression for f, but the analysis yields an explicit bound of the form f(k) = c·k·log k for some constant c that depends only on the combinatorial arguments used. This bound improves on the trivial exponential bound that follows from the general fact that any graph with ω = k can be colored with at most 2^k colors.

The proof proceeds by exploiting the ordering induced by the vertical line L. Every curve meets L at a unique point; sorting the curves by the y‑coordinate of this intersection yields a linear order. The authors split the graph into two subgraphs: the “left” subgraph consisting of the portions of the curves to the left of L, and the “right” subgraph consisting of the portions to the right of L. Because each curve is x‑monotone, the left portions behave like intervals on a line, and classical results on interval graphs (e.g., Dilworth’s theorem and the Erdős–Szekeres chain–antichain decomposition) give a linear bound on the chromatic number of the left subgraph in terms of its clique number.

The right subgraph is more intricate. The authors introduce a “grid‑compression” technique: they overlay a fine rectangular grid on the half‑plane to the right of L and show that each grid cell can contain at most a bounded number of curve segments without creating a large clique. By carefully analyzing the possible crossing patterns (so‑called “switch patterns”) inside the grid, they prove that any large independent set in the right subgraph forces the existence of a relatively small clique, which in turn yields a logarithmic factor in the final bound. The two colorings are then merged, giving the overall bound χ(G) ≤ c·ω(G)·log ω(G).

Two immediate corollaries follow. First, the intersection graphs of rays (half‑infinite straight lines) are χ‑bounded. A ray can be viewed as an x‑monotone curve that starts at a point on L and extends infinitely in one direction; therefore the same argument applies directly. Second, the intersection graphs of unit‑length line segments are χ‑bounded. By rotating and translating each unit segment, one can embed it into an x‑monotone curve intersecting a common vertical line without changing the intersection relationships, and thus the same bound holds.

The paper situates its contributions within the broader literature on χ‑boundedness. Prior work had established χ‑boundedness for interval graphs, comparability graphs, and certain families of geometric intersection graphs such as those defined by axis‑parallel rectangles or convex sets under additional restrictions. The present work extends these results to a new, natural geometric family that had not been covered before. Moreover, the techniques introduced—particularly the combination of order‑based decomposition, grid compression, and control of switch patterns—provide a toolbox that may be adapted to other geometric settings.

In the discussion section the authors outline several promising directions for future research. One line of inquiry is whether the requirement that all curves intersect a single vertical line can be relaxed, perhaps to the condition that the curves intersect a bounded number of vertical lines, while still preserving χ‑boundedness. Another question concerns improving the bound f(k); for instance, can the logarithmic factor be eliminated for special subclasses (e.g., when the curves have bounded curvature or are straight line segments)? Finally, the authors suggest extending the analysis to higher dimensions, where one might consider x‑monotone surfaces intersecting a common plane, and investigate whether analogous χ‑boundedness results hold.

Overall, the paper delivers a substantial advance in the theory of geometric intersection graphs by proving χ‑boundedness for a broad class of x‑monotone curves, and by deriving concrete corollaries for rays and unit segments that have immediate relevance to computational geometry and graph coloring algorithms.