The step Sidorenko property and non-norming edge-transitive graphs

TL;DR

This paper introduces the step Sidorenko property, a stronger version of Sidorenko's Conjecture, and demonstrates that many bipartite graphs lack this property, including a specific edge-transitive graph that is not weakly norming.

Contribution

The paper defines the step Sidorenko property and shows many bipartite graphs do not possess it, providing a counterexample of an edge-transitive graph that is not weakly norming.

Findings

Many bipartite graphs fail the step Sidorenko property

Existence of a bipartite edge-transitive graph that is not weakly norming

Answers a question posed by Hatami

Abstract

Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150].