Two approaches to Sidorenko’s conjecture

Abstract

Sidorenko's conjecture states that for every bipartite graph H on {1, · · · , k} (i,j)∈E(H) h(x i , y j )dµ |V (H)| ≥ h(x, y) dµ 2 |E(H)| 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 [3]. 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.

Finite reflection groups and graph norms

“…Let X, Y , and Z be random variables and suppose that X takes values in a set S, H(X) is the entropy of X, and H(X|Y ) is the conditional entropy of X given Y . Then As in [14,6], we say that a bipartite graph H has Sidorenko's property if H satisfies (5) for all non-negative symmetric functions f , i.e., Sidorenko's conjecture holds for H.…”

Some advances on Sidorenko's conjecture

“…We say that a graph has Sidorenko's property if holds for all graphs . While Sidorenko himself noted that the conjecture holds for some simple graphs such as trees, even cycles and complete bipartite graphs, a spate of recent work , some of which we will describe below, has greatly expanded the class of graphs known to have Sidorenko's property. In this paper, we further this progress, providing three more families of graphs that satisfy the conjecture.…”

The Exact Minimum Number of Triangles in Graphs With Given Order and Size

“…A famous conjecture of Sidorenko [38] asserts, roughly speaking, that the minimal number of F -subgraphs is asymptotically attained by a random graph (we do not give a precise statement of the conjecture here). The conjecture is known to be true for trees, cycles, complete bipartite graphs, 'strongly tree-decomposable graphs' and others, see [2,3,15,21,23,40].…”