No abstract

We consider the problem of determining the possible orders for k-regular, k-connected and bipancyclic subgraphs of the hypercube Qn. For k = 2 and k = 3, the solution to the problem is known. In this paper, we solve the problem for k = 4 by proving that Qn has a 4-regular, 4-connected and bipancyclic subgraph on l vertices if and only if l = 16 or l is an even integer such that 24 ≤ l ≤ 2 n . Further, by improving a result of Ramras, we prove that a k-regular subgraph of Qn is either isomorphic to Q k or has at least 2 k + 2 k−1 vertices. We also improve a result of Mane and Waphare regarding the existence of a k-regular, k-connected and bipancyclic subgraph of Qn. Some applications of our results are given.

No abstract

Let G be a graph obtained by taking the Cartesian product of finitely many cycles. It is known that G is bipancyclic, that is, G contains cycles of every even length from 4 to |V (G)|. We extend this result for the existence of 3-regular subgraphs in G. We prove that G contains a 3-regular, 2-connected subgraph with l vertices if l = 8 or l = 12 or l is an even integer with 16 ≤ l ≤ |V (G)|. For l ∈ {6, 10, 14}, we give necessary and sufficient conditions for the existence of such subgraphs in G. c

In this paper, we consider the problem of decomposing the edge set of the hypercube Qn into two spanning, regular, connected, bipancyclic subgraphs. We prove that if n = n1 + n2 with n1 ≥ 2 and n2 ≥ 2, then the edge set of Qn can be decomposed into two spanning, bipancyclic subgraphs H1 and H2 such that Hi is ni-regular and ni-connected for i = 1, 2.