## Abstract

The Gyárfás–Sumner conjecture says that for every forest (Formula presented.), there is a function (Formula presented.) such that if (Formula presented.) is (Formula presented.) -free then (Formula presented.) (where (Formula presented.) are the chromatic number and the clique number of (Formula presented.)). Louis Esperet conjectured that, whenever such a statement holds, (Formula presented.) can be chosen to be a polynomial. The Gyárfás–Sumner conjecture is only known to be true for a modest set of forests (Formula presented.), and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when (Formula presented.) is a five-vertex path. Here we prove Esperet's conjecture when each component of (Formula presented.) is a star.

Original language | English (US) |
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Pages (from-to) | 318-322 |

Number of pages | 5 |

Journal | Journal of Graph Theory |

Volume | 101 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2022 |

## All Science Journal Classification (ASJC) codes

- Geometry and Topology
- Discrete Mathematics and Combinatorics

## Keywords

- Gyárfás–Sumner conjecture
- chromatic number
- colouring
- induced subgraph
- χ-boundedness