A graph class is \(\chi\)-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erdős conjectured that intersection graphs of straight-line segments in the plane are \(\chi\)-bounded, but this was disproved by Pawlik et al. (2014), who showed another way to force large chromatic number in this class\(\unicode{x2014}\)by triangle-free graphs \(B_k\) with \(\chi(B_k) = k\) constructed by Burling (1965). This also disproved the celebrated conjecture of Scott (1997) that classes of graphs excluding induced subdivisions of a fixed graph are \(\chi\)-bounded.

In this paper, we consider the class 𝒞^d of sphere intersection graphs in R^d for d ≥ 2. We show that for each integer t, the class of all graphs in 𝒞^d that exclude K_{t,t} as a subgraph has strongly sublinear separators. We also prove that 𝒞^d has asymptotic dimension at most 2d+2.

Abstract 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 $$\omega $$ ω has chromatic number at most $$17\omega ^4$$ 17 ω 4 . This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially $$\chi $$ χ -bounded. We also survey $$\chi $$ χ -boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3⋅4^{ω-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time.