Sidorenko's conjecture for a class of graphs: an exposition

TL;DR

This paper provides a clear, self-contained proof of Sidorenko's conjecture for bipartite graphs where one part has a universal vertex, demonstrating the minimal number of subgraph copies in random graphs.

Contribution

It offers a simplified, pedagogical proof of Sidorenko's conjecture for a specific class of bipartite graphs, enhancing understanding and teaching of the conjecture.

Findings

Proof confirms the conjecture for graphs with a universal vertex in one part.

Demonstrates the minimal number of subgraph copies in random graphs.

Provides an accessible proof suitable for educational purposes.

Abstract

A famous conjecture of Sidorenko and Erd\H{o}s-Simonovits states that if H is a bipartite graph then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. The goal of this expository note is to give a short self-contained proof (suitable for teaching in class) of the conjecture if H has a vertex complete to all vertices in the other part.