## Abstract

A celebrated conjecture of Sidorenko and Erdos–Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion. Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A⋃B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A⋃B, there is a positive integer p such that the blow-up H^{p} _{A} formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture. Another way of viewing this latter result is that for every bipartite H there is a positive integer p such that an Lp-version of Sidorenko’s conjecture holds for H.

Original language | English |
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Article number | 2 |

Journal | Discrete Analysis |

Volume | 2021 |

DOIs | |

Publication status | Published - 2021 |

### Bibliographical note

Funding Information:*Supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. †Supported by ERC Consolidator Grant PEPCo 724903 and ERC Starting Grant 676632.

Publisher Copyright:

© 2021. David Conlon, Joonkyung Lee

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Geometry and Topology
- Discrete Mathematics and Combinatorics