On tripartite common graphs

TL;DR

This paper introduces new classes of tripartite common graphs, including triangle-trees and graphs formed by attaching small trees or apex vertices, expanding the known examples of common graphs in graph theory.

Contribution

The authors identify and prove new classes of tripartite common graphs, including triangle-trees and graphs formed by attaching small trees or apex vertices, answering longstanding questions.

Findings

Triangle-trees are common graphs, generalizing Sidorenko's theorems.

For any tree T, a corresponding triangle-tree exists that remains common when T is attached.

Adding many apex vertices to small bipartite graphs yields common graphs.

Abstract

A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdos, conjectured that every graph is common. The conjectures by Erdos and by Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples for common graphs had not seen much progress since then, although very recently, a few more graphs are verified to be common by the flag algebra method or the recent progress on Sidorenko's conjecture. Our contribution here is to give a new class of tripartite common graphs. The first example class is so-called triangle-trees, which generalises two theorems by Sidorenko and answers a question by Jagger, \v{S}\v{t}ov\'i\v{c}ek, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree T, there exists a triangle-tree such that the graph obtained by adding T as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most five vertices give a common graph.