Year: 2025
Type: article
Abstract: When H is a forest, the Gyárfás-Sumner conjecture implies that every graph G with no induced subgraph isomorphic to H and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph G has a stable set of size $$|G|^{1-o(1)}$$ | G | 1 - o ( 1 ) . If H is not a forest, there need not be such a stable set. Second, we prove that when H is a “multibroom”, there is a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a “fractional colouring” version of the Gyárfás-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.
Source: COMBINATORICA
Authors: Tung Nguyen, Alex Scott, Paul Seymour
Institutions: Princeton University, University of Oxford
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4414577375