Some advances on Sidorenko's conjecture

Abstract

A bipartite graph H is said to have Sidorenko's property if the probability that the uniform random mapping from V (H) to the vertex set of any graph G is a homomorphism is at least the product over all edges in H of the probability that the edge is mapped to an edge of G. In this paper, we provide three distinct families of bipartite graphs that have Sidorenko's property. First, using branching random walks, we develop an embedding algorithm which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko's property. Second, we use the concept of locally dense graphs to prove that subdivisions of certain graphs, including cliques, have Sidorenko's property. Third, we prove that if H has Sidorenko's property, then the Cartesian product of H with an even cycle also has Sidorenko's property.

Finite reflection groups and graph norms

“…where χ L is the colouring obtained by doubling the 'left half' of the rainbow cycle, represented by the vector (1, 2, 3, 3, 2, 1). A similar equation also holds for h, namely, h 2 = F; χ R C 6 , where χ R = (6,5,4,4,5,6) is the colouring obtained by doubling the 'right half' of the rainbow cycle. We have therefore bounded F; χ C 6 from above by the geometric mean of two functions of the same form, but simpler in the sense that they both contain fewer colours.…”

“…Let us mention another example of a norming graph. Following [6], we say that a graph is a K 2,t -replacement of H if each edge of H is replaced with a copy of K 2,t by identifying the two vertices of the edge with the two vertices on the smaller side of K 2,t . Finally, we remark that one may presumably use the exceptional reflection groups E 6 , E 7 , and E 8 to build some more exotic (weakly) norming graphs, though we have not pursued this further.…”

“…That is, when H is weakly norming, H satisfies Sidorenko's conjecture [22,23], which says exactly that for any bipartite graph H and any bounded measurable function f , an inequality of the form (5) holds. Sidorenko's conjecture is one of the major open problems in extremal graph theory, and there has been much recent work [4,6,14,16,24] verifying the conjecture for a widening class of graphs.…”

“…However, this is not the only application of our results to Sidorenko's conjecture. By applying the entropy techniques developed in [16,14,24,6], we will prove that weakly norming graphs can also be used as building blocks for constructing new graphs that satisfy the conjecture. We refer the reader to Section 5.3 for more details.…”

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

“…Combining this with Claim 8.11 (2), we see that again A k = R k ∪X ⊆ A * k . Then Claim 8.11 (3) implies that for all i ∈ [k−1] we have A * σ(i) \A i ⊆ A k ⊆ A * k and so A * σ(i) ⊆ A i . By Claim 8.11 (4), there is j…”