Panconnectivity and edge-pancyclicity of 3-ary N-cubes

Abstract

In this article, we study some topological properties of k -ary n-cubes Q k n . We show that each edge in Q k n lies on a cycle of every length from k to k n . We also show that Q k n is both bipanconnected and edge-bipancyclic, where n ≥ 2 is an integer and k ≥ 2 is an even integer.

Bipanconnectivity and Bipancyclicity in k-ary n-cubes

“…We also strengthen the results in [15] and [27] by introducing a path-shortening technique, called progressive shortening, and show that the construction of paths using this technique enables us to efficiently construct paths in a distributed fashion and so solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q k n , even in the presence of a faulty processor (even in Q k 2 , the solution to this problem is not possible using the paths constructed in [27]). …”

“…Note that putting k ¼ 3 in Theorem 6 yields the result from [15] that Q 3 n is edge-pancyclic, and also resolves the question for arbitrary k, as was posed in [15]. The following corollary is immediate, given the fact that the diameter of Q k n , when k is odd, is , and ðk À 1Þ-pancyclic.…”

“…3, is bipanconnected or bipancyclic was left as an open question. However, in relation to this question, it was proven in [15] that Q 3 n is edge-pancyclic, for all n ! 2.…”

“…With regard to k-ary n-cubes, these notions have been considered in [15] and [27]. In particular, it was proven in [27] that Q k 2 is almostHamiltonian connected, bipanconnected, and bipancyclic; that Q k n is almost-Hamiltonian connected, for any k; and that Q k n is Hamiltonian-connected, for odd k. Recently, it has been proven in [15] that Q 3 n is edge-pancyclic.…”

“…With regard to k-ary n-cubes, these notions have been considered in [15] and [27]. In particular, it was proven in [27] that Q k 2 is almostHamiltonian connected, bipanconnected, and bipancyclic; that Q k n is almost-Hamiltonian connected, for any k; and that Q k n is Hamiltonian-connected, for odd k. Recently, it has been proven in [15] that Q 3 n is edge-pancyclic. It was posed as an open problem in [27] as to whether their results on bipanconnectivity and bipancyclicity for Q k 2 could be extended to Q k n , for arbitrary n, and it was posed as an open problem in [15] as to whether their results on panconnectivity and pancyclicity could be extended to Q k n , for arbitrary k. In this paper, we provide precise answers to both these questions.…”