On the Tur\'an Density of $C_{10}$ in the Hypercube

Abstract

The $n$-dimensional hypercube $Q_n$ is the graph with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. For a graph $H$, let $\operatorname{ex}(Q_n,H)$ be the maximum number of edges in an $H$-free subgraph of $Q_n$. The hypercube Tur\'an density of $H$ is defined by $\pi_{\square}(H)=\lim_{n\rightarrow\infty}\operatorname{ex}(Q_n,H)/|E(Q_n)|$. In this note, we prove \[ \frac{1}{8} \leq \pi_{\square}(C_{10}) \leq 0.36577. \] For the upper bound, we prove $\pi_{\square}(C_{10}) \leq \pi_{\square}(C_6)$, which, together with a result of Baber, gives the stated upper bound. For the lower bound, we prove that $\operatorname{ex}(Q_n,C_{10})>|E(Q_n)|/8$ for every $n \geq 2$.