Two approaches to Sidorenko’s conjecture

Abstract: Sidorenko’s conjecture states that the number of homomorphisms from a bipartite graph H to a graph G is at least the expected number of homomorphisms from H to the binomial random graph with the same expected edge density as G. In this talk, I will present two approaches to the conjecture. First, I will 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, and 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 a1, a2 in A such that each vertex a ∈ A satisfies N(a) ⊆ N(a1) or N(a) ⊆ N(a2). Second, I will prove that if T is a tree and H is a bipartite graph satisfying Sidorenko’s conjecture, then the Cartesian product of T and H also satisfies idorenko’s conjecture. This result implies that, for all d ≥ 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko’s conjecture. Joint w/ Jeong Han Kim (KIAS) and Joonkyung Lee (Oxford)