Grounded L-graphs are polynomially $\chi$-bounded

TL;DR

This paper proves that grounded L-graphs have a polynomial upper bound on their chromatic number relative to their clique number, improving previous bounds and extending results to circle graphs.

Contribution

It establishes a polynomial $ ext{chi}$-boundedness result for grounded L-graphs, advancing understanding of their coloring properties.

Findings

Grounded L-graphs are polynomially $ ext{chi}$-bounded with bound $17 ext{omega}^4$.

Improves previous doubly-exponential bounds for these graphs.

Provides a survey of recent techniques in $ ext{chi}$-boundedness for geometric intersection graphs.

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 $\omega$ 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.