## Abstract

The classical Hadwiger conjecture dating back to 1940s states that any graph of chromatic number at least r has the clique of order r as a minor. Hadwiger's conjecture is an example of a well-studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on n vertices of independence number (Formula presented.) at most r. If true Hadwiger's conjecture would imply the existence of a clique minor of order (Formula presented.). Results of Kühn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that G is H-free for some bipartite graph H then one can find a polynomially larger clique minor. This has recently been extended to triangle-free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph H, answering a question of Dvořák and Yepremyan. In particular, we show that any (Formula presented.) -free graph has a clique minor of order (Formula presented.), for some constant (Formula presented.) depending only on s. The exponent in this result is tight up to a constant factor in front of the (Formula presented.) term.

Original language | English (US) |
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Pages (from-to) | 327-338 |

Number of pages | 12 |

Journal | Random Structures and Algorithms |

Volume | 60 |

Issue number | 3 |

DOIs | |

State | Published - May 2022 |

## All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

## Keywords

- H-free graphs
- Hadwiger conjecture
- graph minor
- independent set expansion