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Polynomial Gyárfás-Sumner conjecture for graphs of bounded boxicity

James Davies, Yelena Yuditsky

TL;DR

For every $d$ and forest $F$, the authors prove that the class of box-intersection graphs in $\mathbb{R}^d$ with no induced $F$ subgraph is polynomially $\chi$-bounded. The method combines a directed-graph framework with $(k,m)$-gradings and calm copies to find path-induced copies of rooted trees, together with a decomposition by intersection patterns $\mathcal{P}_d$ to control chromatic number. They derive an explicit bound $\chi(G)\le (2rk^d\omega(G)^2)^{4^{d}}$ for graphs avoiding an induced $T_{r,k}$, by showing large chi implies a path-induced tree that leads to a contradiction. This work extends the Gyárfás–Sumner program to high-dimensional geometric intersection graphs, connects to Erdős–Hajnal properties, and identifies several open problems and a Pollyanna-type phenomenon for box graphs.

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) $χ$-bounded.

Polynomial Gyárfás-Sumner conjecture for graphs of bounded boxicity

TL;DR

For every and forest , the authors prove that the class of box-intersection graphs in with no induced subgraph is polynomially -bounded. The method combines a directed-graph framework with -gradings and calm copies to find path-induced copies of rooted trees, together with a decomposition by intersection patterns to control chromatic number. They derive an explicit bound for graphs avoiding an induced , by showing large chi implies a path-induced tree that leads to a contradiction. This work extends the Gyárfás–Sumner program to high-dimensional geometric intersection graphs, connects to Erdős–Hajnal properties, and identifies several open problems and a Pollyanna-type phenomenon for box graphs.

Abstract

We prove that for every positive integer and forest , the class of intersection graphs of axis-aligned boxes in with no induced subgraph is (polynomially) -bounded.
Paper Structure (4 sections, 7 theorems, 2 equations)

This paper contains 4 sections, 7 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Directed graphs
  3. Box intersection graphs
  4. Open problems

Key Result

Theorem 1

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

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 4 more