Embedding of Cycles in Twisted Cubes with Edge-Pancyclic

In this paper, we study the embedding of cycles in twisted cubes. It has been proven in the literature that, for any integer l, 4≤l≤2 ⁿ , a cycle of length l can be embedded with dilation 1 in an n-dimensional twisted cube, n≥3. We obtain a stronger result of embedding of cycles with edge-pancyclic. We prove that, for any integer l, 4≤l≤2 ⁿ , and a given edge (x,y) in an n-dimensional twisted cube, n≥3, a cycle C of length l can be embedded with dilation 1 in the n-dimensional twisted cube such that (x,y) is in C in the twisted cube. Based on the proof of the edge-pancyclicity of twisted cubes, we further provide an O(llog l+n ²+nl) algorithm to find a cycle C of length l that contains (u,v) in TQ _n for any (u,v)∈E(TQ _n ) and any integer l with 4≤l≤2 ⁿ .     

Panconnectivity and edge-pancyclicity of 3-ary <Emphasis Type="Italic">N</Emphasis>-cubes

We study two topological properties of the 3-ary n-cube Q _n ³. Given two arbitrary distinct nodes x and y in Q _n ³, we prove that there exists an x–y path of every length ranging from d(x,y) to 3 ⁿ −1, where d(x,y) is the length of a shortest path between x and y. Based on this result, we prove that Q _n ³ is edge-pancyclic by showing that every edge in Q _n ³ lies on a cycle of every length ranging from 3 to 3 ⁿ .

Node-pancyclicity and edge-pancyclicity of crossed cubes

Crossed cubes are important variants of the hypercubes. It has been proven that crossed cubes have attractive properties in Hamiltonian connectivity and pancyclicity. In this paper, we study two stronger features of crossed cubes. We prove that the n-dimensional crossed cube is not only node-pancyclic but also edge-pancyclic for n?2. We also show that the similar property holds for hypercubes.     

Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs

A bipartite graph G is bipancyclic if G has a cycle of length l for every even 4?l?|V(G)|. For a bipancyclic graph G and any edge e, G is edge-bipancyclic if e lies on a cycle of any even length l of G. In this paper, we show that the bubble-sort graph Bn is bipancyclic for n?4 and also show that it is edge-bipancyclic for n?5. Assume that F is a subset of E(Bn). We prove that Bn−F is bipancyclic, when n?4 and |F|?n−3. Since Bn is a (n−1)-regular graph, this result is optimal in the worst case.     

Edge-pancyclicity and path-embeddability of bijective connection graphs

An n-dimensional Bijective Connection graph (in brief BC graph) is a regular graph with 2ⁿ nodes and n2ⁿ⁻¹ edges. The n-dimensional hypercube, crossed cube, Möbius cube, etc. are some examples of the n-dimensional BC graphs. In this paper, we propose a general method to study the edge-pancyclicity and path-embeddability of the BC graphs. First, we prove that a path of length l with dist(X_n, x, y) + 2 ? l ? 2ⁿ − 1 can be embedded between x and y with dilation 1 in X_n for x, y ∈ V(X_n) with x ≠ y in X_n, where X_n (n ? 4) is a n-dimensional BC graph satisfying the three specific conditions and V(X_n) is the node set of X_n. Furthermore, by this result, we can claim that X_n is edge-pancyclic. Lastly, we show that these results can be applied to not only crossed cubes and Möbius cubes, but also other BC graphs except crossed cubes and Möbius cubes. So far, the research on edge-pancyclicity and path-embeddability has been limited in some specific interconnection architectures such as crossed cubes, Möbius cubes.     

Edge-pancyclicity of Möbius cubes

The Möbius cube Mn is a variant of the hypercube Qn and has better properties than Qn with the same number of links and processors. It has been shown by Fan [J. Fan, Hamilton-connectivity and cycle-embedding of Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117] and Huang et al. [W.-T. Huang, W.-K. Chen, C.-H. Chen, Pancyclicity of Möbius cubes, in: Proc. 9th Internat. Conf. on Parallel and Distributed Systems (ICPADS'02), 17-20 Dec. 2002, pp. 591-596], independently, that Mn contains a cycle of every length from 4 to n². In this paper, we improve this result by showing that every edge of Mn lies on a cycle of every length from 4 to n² inclusive.