Layered subgraphs of the hypercube

TL;DR

This paper demonstrates that there are large girth subgraphs of the hypercube that are not layered, even when induced, answering a previously open question.

Contribution

It proves the existence of non-layered, arbitrarily large girth subgraphs of the hypercube, including induced subgraphs, addressing an open problem.

Findings

Existence of non-layered hypercube subgraphs with arbitrarily large girth

Such subgraphs can be induced

Answers a question posed by Axenovich, Martin, and Winter

Abstract

A subgraph of the $n$-dimensional hypercube is called 'layered' if it is a subgraph of a layer of some hypercube. In this paper we show that there exist subgraphs of the cube of arbitrarily large girth that are not layered. This answers a question of Axenovich, Martin and Winter. Perhaps surprisingly, these subgraphs may even be taken to be induced.