Polynomial Gy\'arf\'as-Sumner conjecture for graphs of bounded boxicity

TL;DR

This paper proves that for any fixed dimension and forest, the class of intersection graphs of axis-aligned boxes avoiding that forest as an induced subgraph has chromatic number bounded by a polynomial function of its clique number.

Contribution

It establishes a polynomial $ ext{chi}$-boundedness result for intersection graphs of boxes in fixed dimensions excluding a forest as an induced subgraph.

Findings

Polynomial $ ext{chi}$-boundedness for box intersection graphs

Bounded boxicity graphs exclude certain induced forests

Extension of Gyárfás-Sumner conjecture to higher dimensions

Abstract

We prove that for every positive integer $d$ and forest $F$, the class of intersection graphs of axis-aligned boxes in $\mathbb{R}^d$ with no induced $F$ subgraph is (polynomially) $\chi$-bounded.