Extremal numbers for cycles in a hypercube

TL;DR

This paper establishes an upper bound on the maximum number of edges in subgraphs of hypercubes that avoid certain cycle subgraphs, advancing understanding of extremal combinatorics in hypercube graphs.

Contribution

It provides a new asymptotic upper bound for extremal numbers of specific even cycles in hypercube graphs, generalizing previous results.

Findings

Derived an explicit upper bound for $ex(Q_n, C_{4k+2})$

Extended extremal graph theory to hypercube structures

Improved understanding of cycle avoidance in high-dimensional graphs

Abstract

Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} + \frac{1}{3(2k-2)}} 2^n).$$