Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube

TL;DR

This paper adapts Razborov's flag algebra method to the hypercube, establishing new upper bounds on the maximum edges in 4- and 6-cycle-free subgraphs, and analyzing vertex densities in layered hypercube subgraphs.

Contribution

It introduces a modified flag algebra approach for hypercubes and provides improved bounds on cycle-free subgraph sizes and vertex densities.

Findings

Maximum edges in 4-cycle-free subgraphs ≤ 0.6068 times total edges

Maximum edges in 6-cycle-free subgraphs ≤ 0.3755 times total edges

Vertex density of three middle layers without 4-cycles ≤ 2.15121 times n choose n/2

Abstract

In this paper we modify slightly Razborov's flag algebra machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges of a 4-cycle-free subgraph of the n-dimensional hypercube is at most 0.6068 times the number of its edges. We also improve the upper bound on the number of edges for 6-cycle-free subgraphs of the n-dimensional hypercube from the square root of 2 - 1 to 0.3755 times the number of its edges. Additionally, we show that if the n-dimensional hypercube is considered as a poset, then the maximum vertex density of three middle layers in an induced subgraph without 4-cycles is at most 2.15121 times n choose n/2.