Induced subgraphs of hypercubes

TL;DR

This paper uses combinatorial methods to analyze the structure of induced subgraphs in hypercubes, focusing on the maximum number of full vertices and edge coverage by pairs of subgraphs.

Contribution

It applies the Kruskal-Katona Theorem to determine maximum full vertices in induced subgraphs and explores optimal edge coverage by two subgraphs in hypercubes.

Findings

Maximum number of full vertices in induced subgraphs computed

Optimal partitioning of hypercube edges into two induced subgraphs identified

Method provides bounds for subgraph sizes covering all hypercube edges

Abstract

Let $Q_k$ denote the $k$-dimensional hypercube on $2^k$ vertices. A vertex in a subgraph of $Q_k$ is {\em full} if its degree is $k$. We apply the Kruskal-Katona Theorem to compute the maximum number of full vertices an induced subgraph on $n\leq 2^k$ vertices of $Q_k$ can have, as a function of $k$ and $n$. This is then used to determine $\min(\max(|V(H_1)|, |V(H_2)|))$ where (i) $H_1$ and $H_2$ are induced subgraphs of $Q_k$, and (ii) together they cover all the edges of $Q_k$, that is $E(H_1)\cup E(H_2) = E(Q_k)$.