5-Regular Subgraphs in Hypercubes

One of the fundamental properties of the hypercube \ (Qₙ \) is that it is bipancyclic as \ (Qₙ \) has a cycle of length \ (l \) for every even integer \ (l \) with \ (4 l 2ⁿ \). 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 \ (Qₙ \) 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 \ (Qₙ \) 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 2ⁿ \). For general \ (k \), we establish that every \ (k \) -regular subgraph of \ (Qₙ \) has \ (2ᵏ, 2ᵏ + 2^k-1 \) or at least \ (2ᵏ + 2^k-1 + 2^k-3 \) vertices.