Maximum density of vertex-induced perfect cycles and paths in the hypercube

TL;DR

This paper investigates the maximum density of specific perfect cycles and paths within hypercubes, providing exact limits for certain cases and conjectures for larger dimensions, linking combinatorial configurations with graph inducibility.

Contribution

It determines exact asymptotic densities for perfect cycles and paths in hypercubes and introduces conjectures for higher dimensions, connecting these problems with graph inducibility.

Findings

Calculated (H,4) and \u03c0(P,3) for specific configurations

Established connections with sequence counting and graph inducibility

Proposed conjectures for (C_{2d},d) and (P_{d+1},d)

Abstract

Let $H$ and $K$ be subsets of the vertex set $V(Q_d)$ of the $d$-cube $Q_d$ (we call $H$ and $K$ configurations in $Q_d$). We say $K$ is an \emph{exact copy} of $H$ if there is an automorphism of $Q_d$ which sends $H$ to $K$. If $d$ is a positive integer and $H$ is a configuration in $Q_d$, we define $\pi(H,d)$ to be the limit as $n$ goes to infinity of the maximum fraction, over all subsets $S$ of $V(Q_n)$, of sub-$d$-cubes of $Q_n$ whose intersection with $S$ is an exact copy of $H$. We determine $\pi(C_8,4)$ and $\pi(P_4,3)$ where $C_8$ is a "perfect" 8-cycle in $Q_4$ and $P_4$ is a "perfect" path with 4 vertices in $Q_3$, and make conjectures about $\pi(C_{2d},d)$ and $\pi(P_{d+1},d)$ for larger values of $d$. In our proofs there are connections with counting the number of sequences with certain properties and with the inducibility of certain small graphs. In particular, we needed to determine the inducibility of two vertex disjoint edges in the family of bipartite graphs.