A class of graphs of zero Tur\'an density in a hypercube

TL;DR

This paper investigates which cubical graphs have zero Turán density in hypercubes, disproving a previous conjecture by providing counterexamples and establishing new conditions for zero Turán density.

Contribution

It demonstrates that having partite representation is not necessary for zero Turán density, and introduces broader criteria for zero Turán density in cubical graphs.

Findings

Counterexamples of graphs with zero Turán density but no partite representation.

Graphs with all blocks having partite representation have zero Turán density.

Disproved the conjecture that partite representation characterizes zero Turán density.

Abstract

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is at least a positive proportion of the number of edges in $Q_n$, $H$ is said to have a positive Tur\'an density in a hypercube or simply a positive Tur\'an density; otherwise it has a zero Tur\'an density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has a positive or a zero Tur\'an density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of cubical graphs, ones having so-called partite representation, that have a zero Tur\'an density. He raised a question whether this gives a characterisation, i.e., whether a cubical graph has zero Tur\'an density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of cubical graphs which have no partite representation, but on the other hand, have a zero Tur\'an density. In addition, we show that any graph whose every block has partite representation has a zero Tur\'an density in a hypercube.