5-Regular Subgraphs in Hypercubes

One of the fundamental properties of the hypercube \( Q_n \) is that it is bipancyclic as \( Q_n \) has a cycle of length \( l \) for every even integer \( l \) with \( 4 ≤ l ≤ 2^n \). We consider the following problem of generalizing this property: For a given integer \( k \) with \( 3 ≤ k ≤ n \), determine all integers \( l \) for which there exists an \( l \)-vertex, \( k \)-regular subgraph of \( Q_n \) that is both \( k \)-connected and bipancyclic. The solution to this problem is known for \( k = 3 \) and \( k = 4 \). In this paper, we solve the problem for \( k = 5 \). We prove that \( Q_n \) contains a \( 5 \)-regular subgraph on \( l \) vertices that is both \( 5 \)-connected and bipancyclic if and only if \( l ∈ \{32, 48\} \) or \( l \) is an even integer satisfying \( 52 ≤ l ≤ 2^n \). For general \( k \), we establish that every \( k \)-regular subgraph of \( Q_n \) has \( 2^k, 2^k + 2ᵏ⁻¹ \) or at least \( 2^k + 2ᵏ⁻¹ + 2ᵏ⁻³ \) vertices.