Grounded L-Graphs Are Polynomially $$\chi $$ -Bounded

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.