## Abstract

Sidorenko’s conjecture states that for every bipartite graph H on {1,· ·, k} (Formula Presented) holds, where μ is the Lebesgue measure on [0, 1] and h is a bounded, nonnegative, symmetric, measurable function on [0, 1]^{2}. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph H to a graph G is asymptotically at least the expected number of homomorphisms from H to the Erdős-Rényi random graph with the same expected edge density as G. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph H with bipartition A ∪ B is tree-arrangeable if neighborhoods of vertices in A have a certain tree-like structure. We show that Sidorenko’s conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko’s conjecture holds if there are two vertices a_{1}, a_{2} in A such that each vertex a ∈ A satisfies N(a) ⊆ N(a_{1}) or N(a) ⊆ N(a_{2}), and also implies a recent result of Conlon, Fox, and Sudakov (2010). Second, if T is a tree and H is a bipartite graph satisfying Sidorenko’s conjecture, then it is shown that the Cartesian product T □ H of T and H also satisfies Sidorenko’s conjecture. This result implies that, for all d ≥ 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko’s conjecture.

Original language | English |
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Pages (from-to) | 5057-5074 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2016 Jul |

### Bibliographical note

Funding Information:The first author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2012R1A2A2A01018585) and KIAS internal Research Fund CG046001. This work was partially carried out while the author was visiting Microsoft Research, Redmond, and Microsoft Research, New England. The third author was supported by ILJU Foundation of Education and Culture.

Publisher Copyright:

© 2015 American Mathematical Society.

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics