Tung-Yang Ho

Component connectivity of the hypercubes

The r-component connectivity κ r (G) of the non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. So, κ2 is the usual connectivity. In this paper, we determine the r-component connectivity of the hypercube Q n for r=2, 3, …, n+1, and we classify all the corresponding optimal solutions.

On mutually independent hamiltonian paths

Abstract Let P 1 = 〈 v 1 , v 2 , v 3 , … , v n 〉 and P 2 = 〈 u 1 , u 2 , u 3 , … , u n 〉 be two hamiltonian paths of G . We say that P 1 and P 2 are independent if u 1 = v 1 , u n = v n , and u i ≠ v i for 1 i n . We say a set of hamiltonian paths P 1 , P 2 , … , P s of G between two distinct vertices are mutually independent if any two distinct paths in the set are independent. We use n to denote the number of vertices and use e to denote the number of edges in graph G . Moreover, we use e to denote the number of edges in the complement of G . Suppose that G is a graph with e ≤ n − 4 and n ≥ 4 . We prove that there are at least n − 2 − e mutually independent hamiltonian paths between any pair of distinct vertices of G except n = 5 and e = 1 . Assume that G is a graph with the degree sum of any two non-adjacent vertices being at least n + 2 . Let u and v be any two distinct vertices of G . We prove that there are deg G ( u ) + deg G ( v ) − n mutually independent hamiltonian paths between u and v if ( u , v ) ∈ E ( G ) and there are deg G ( u ) + deg G ( v ) − n + 2 mutually independent hamiltonian paths between u and v if otherwise.

Fault-tolerant hamiltonicity and fault-tolerant hamiltonian connectivity of the folded Petersen cube networks

Some research on the folded Petersen cube networks have been published for the past several years due to its favourite properties. In this paper, we consider the fault-tolerant hamiltonicity and the fault-tolerant hamiltonian connectivity of the folded Petersen cube networks. We use FPQ n, k to denote the folded Petersen cube networks of parameters n and k. In this paper, we show that FPQ n, k −F remains hamiltonian for any F ⊆ V(FPQ n, k )∪E(FPQ n, k ) with |F|≤n+3k−2 and FPQ n, k −F remains hamiltonian connected for any F ⊆ V(FPQ n, k )∪E(FPQ n, k ) with |F|≤n+3k−3 if (n, k)∉{(0, 1)}∪{(n, 0) | n is a positive integer}. Moreover, this result is optimal.

On the extremal number of edges in hamiltonian connected graphs

Abstract Assume that n and δ are positive integers with 3 ≤ δ n . Let h c ( n , δ ) be the minimum number of edges required to guarantee an n -vertex graph G with minimum degree δ ( G ) ≥ δ to be hamiltonian connected. Any n -vertex graph G with δ ( G ) ≥ δ is hamiltonian connected if | E ( G ) | ≥ h c ( n , δ ) . We prove that h c ( n , δ ) = C ( n − δ + 1 , 2 ) + δ 2 − δ + 1 if δ ≤ ⌊ n + 3 × ( n mod 2 ) 6 ⌋ + 1 , h c ( n , δ ) = C ( n − ⌊ n 2 ⌋ + 1 , 2 ) + ⌊ n 2 ⌋ 2 − ⌊ n 2 ⌋ + 1 if ⌊ n + 3 × ( n mod 2 ) 6 ⌋ + 1 δ ≤ ⌊ n 2 ⌋ , and h c ( n , δ ) = ⌈ n δ 2 ⌉ if δ > ⌊ n 2 ⌋ .

Fault-tolerant hamiltonian connectivity of the WK-recursive networks

Abstract Many research on the WK-recursive network has been published during the past several years due to its favorite properties. In this paper, we consider the fault-tolerant hamiltonian connectivity of the WK-recursive network. We use K ( d , t ) to denote the WK-recursive network of level t , each of which basic modules is a d -vertex complete graph, where d > 1 and t ⩾ 1 . The fault-tolerant hamiltonian connectivity H f κ ( G ) is defined to be the maximum integer k such that G is k fault-tolerant hamiltonian connected if G is hamiltonian connected and is undefined otherwise. In this paper, we prove that H f κ ( K ( d , t ) ) = d - 4 if d ⩾ 4 .

FAULT-TOLERANT HAMILTONIAN LACEABILITY AND FAULT-TOLERANT CONDITIONAL HAMILTONIAN FOR BIPARTITE HYPERCUBE-LIKE NETWORKS

A bipartite graph G is hamiltonian laceable if there is a hamiltonian path between any two vertices of G from distinct vertex bipartite sets. A bipartite graph G is k-edge fault-tolerant hamiltonian laceable if G - F is hamiltonian laceable for every F ⊆ E(G) with |F| ≤ k. A graph G is k-edge fault-tolerant conditional hamiltonian if G - F is hamiltonian for every F ⊆ E(G) with |F| ≤ k and δ(G - F) ≥ 2. Let G0 = (V0, E0) and G1 = (V1, E1) be two disjoint graphs with |V0| = |V1|. Let Er = {(v,ɸ(v)) | v ϵ V0,ɸ(v) ϵ V1, and ɸ: V0 → V1 is a bijection}. Let G = G0 ⊕ G1 = (V0 ⋃ V1, E0 ⋃ E1 ⋃ Er). The set of n-dimensional hypercube-like graphHn is defined recursively as (a) H1 = K2, K2 is the complete graph with two vertices, and (b) if G0 and G1 are in Hn, then G = G0 ⊕ G1 is in Hn+1. Let Bn be the set of graphs G where G is bipartite and G ϵ Hn. In this paper, we show that every graph in Bn is (n - 2)-edge fault-tolerant hamiltonian laceable if n ≥ 2 and every graph in Bn is (2n - 5)-edge fault-tolerant conditional hamiltonian if n ≥ 3.

Fault-Tolerant Hamiltonicity of the WK-Recursive Networks

Many research on the WK-recursive network has been published during the past several years due to its favorite properties. In this paper, we consider the fault-tolerant hamiltonian connectivity of the WK-recursive network. We use K(d,t) to denote the WK-recursive network of level t, each of which basic modules is a d-vertex complete graph. The fault-tolerant hamiltonian connectivity is defined to be the maximum integer k such that G is k fault-tolerant hamiltonian connected if G is hamiltonian connected and is undefined otherwise. In this paper, we prove that the fault-tolerant hamiltonian connectivity of K(d,t) is d-4.

A NEW ISOMORPHIC DEFINITION OF THE CROSSED CUBE AND ITS SPANNING CONNECTIVITY

In this paper, we propose a slightly different definition of the crossed cube. The interconnection network obtained form our new definition is exactly isomorphic to the one obtained from the original definition proposed by Efe. It is known that the crossed cube is not node symmetric. However, using our new definition, it reveals some relative symmetric properties of the crossed cube. We can take advantage of the symmetry to study the spanning connectivity of the crossed cube.