On 4-regular 4-connected bipancyclic subgraphs of hypercubes

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.

Existence of 3-regular subgraphs in Cartesian product of cycles

“…The hypercube Q n is an n-regular, n-connected bipancyclic graph whereas the product of n cycles is a 2n-regular, 2n-connected, bipancyclic or pancyclic graph (see [1,4]). Regular subgraphs, bipancyclicity and connectivity properties of hypercubes and the product of cycles are studied in [1,3,4,[5][6][7][8][9][10][11].…”

On conditional connectivity of the Cartesian product of cycles

“…The following result is analogous to the result of hypercubes which states that if K is a subgraph of the hypercube Q n isomorphic to Q h , then every vertex of Q n which is not in K has at most one neighbour in K; see [1].…”

“…In the following figure, a 2-regular subgraph W 2 2 and a 3-regular subgraph W 2 3 of the graph C 5 C 5 are shown by bold lines. It is known that a smallest h-regular subgraph of the hypercube Q n is isomorphic to Q h (see [1]). We prove the analogous result for the Cartesian product of cycles.…”

On regular subgraphs of augmented cubes

“…This will be useful to get subgraphs of AQ n with less number of vertices which retain the important properties of AQ n such as regularity, pancyclicity and high connectivity. For hypercubes, this problem is studied in [2,3,14].…”

“…Similar results for the classes of the Cartesian product of cycles and the Cartesian product of paths are obtained in [1] and [13], respectively. For the existence of 4-regular subgraphs, Borse and Shaikh [3] established that there exists a 4-regular, 4-connected and bipancyclic subgraph on l vertices in the hypercube Q n if and only if l ¼ 16 or l is an even integer with 24 l 2 n :…”