Gyárfás–Sumner conjecture for arbitrary trees

Prove that the class of graphs excluding any fixed tree T as an induced subgraph is chi-bounded for every tree T, thereby resolving the general Gyárfás–Sumner conjecture.

Background

The paper discusses chi-boundedness for hereditary graph classes defined by forbidden induced subgraphs. The Gyárfás–Sumner conjecture, proposed independently by Gyárfás and Sumner, asserts that for every tree T there is a function bounding the chromatic number of every T-free graph in terms of its clique number.

The paper notes that the conjecture is known for trees of radius two, but remains unresolved for arbitrary trees. This is background cited in the introduction rather than a problem specific to the perfect-divisibility results proved in the paper.

References

This leads to the Gyárfás--Sumner conjecture, proposed independently by Gyárfás and Sumner. It asserts that the class of T-free graphs is χ-bounded for every tree T. Kierstead and Penrice proved the conjecture when T has radius two, but it remains open in general.

— Every fork-free graph is perfectly weight divisible  (2608.13519 - Liu et al., 13 Aug 2026) in Section 1, Introduction

Gyárfás and Sumner conjectured, in 1975 and 1981, respectively, that for every tree $T$, the class of $T$-free graphs is $\chi$-bounded. This conjecture is known only for relatively few classes of forbidden trees~.

— Perfect Divisibility, Linear Divisibility and Chair-Free Graphs  (2608.14519 - Wang et al., 14 Aug 2026) in Section 1, Introduction