D. Frank Hsu

Combinatorial network theory

Preface. 1. Additive Group Theory Applied to Network Topology. 2. Connectivity of Cayley Digraphs. 3. De Bruijn Digraphs, Kautz Digraphs, and Their Generalizations. 4.Link-Connectivities of Extended Double Loop Networks. 5.Dissemination of Information in Interconnection Networks (Broadcasting & Gossiping.)

Fault tolerance properties of pyramid networks

In this paper, we study the pyramid network (also called pyramid), one of the important architectures in parallel computing, network computing, and image processing. Some properties of pyramid networks are investigated. We determine the line connectivity and the fault diameters in pyramid networks. We show how to construct a path between two nodes in the faulty pyramid networks in polynomial time. A polynomial-time algorithm is also given for generating the containers in pyramid networks. Our results show that pyramid networks have very good fault tolerance properties.

On the k -diameter of k -regular k -connected graphs

Abstract We study the k -diameter of k -regular k -connected graphs. Among other results, we show that every k -regular k -connected graph on n vertices has k -diameter at most n /2 and this upper bound cannot be improved when n =4 k −6+ i (2 k −4). In particular, the maximal 3-diameter of 3-regular graphs with 2 n vertices is equal to n .

Short containers in Cayley graphs

The star diameter of a graph measures the minimum distance from any source node to several other target nodes in the graph. For a class of Cayley graphs from abelian groups, a good upper bound for their star diameters is given in terms of the usual diameters and the orders of elements in the generating subsets. This bound is tight for several classes of graphs including hypercubes and directed n-dimensional tori. The technique used is the so-called disjoint ordering for a system of subsets, due to Gao, Novick and Qiu [S. Gao, B. Novick, K. Qiu, From Halls matching theorem to optimal routing on hypercubes, J. Comb. Theory B 74 (1998) 291-301].

On the hardness of counting problems of complete mappings

A complete mapping of an algebraic structure (G,+) is a bijection f(x) of G over G such that f(x)=x+h(x) for some bijection h(x). A question often raised is, given an algebraic structure G, how many complete mappings of G there are. In this paper we investigate a somewhat different problem. That is, how difficult it is to count the number of complete mappings of G. We show that for a closed structure, the counting problem is #P-complete. For a closed structure with a left-identity and left-cancellation law, the counting problem is also #P-complete. For an abelian group, on the other hand, the counting problem is beyond the #P-class. Furthermore, the famous counting problems of n-queen and toroidal n-queen problems are both beyond the #P-class.

De Bruijn Digraphs, Kautz Digraphs, and Their Generalizations

An interesting problem in network designs is as follows: Given natural numbers n and d, find a digraph (directed graph) with n vertices, each of which has outdegree at most d, to minimize the diameter and to maximize the connectivity. This is a multiobjective optimization problem. Usually, for such a problem, solution is selected based on tradeoff between two objective fuctions. However, for this problem, it is different; that is, there exists a solution which is optimal or nearly optimal to both. Such a solution comes from study of de Bruijn digraphs, Kautz digraphs, and their generalizations. In this chapter, we introduce and survey results on these subjects.

The construction of mutually independent Hamiltonian cycles in bubble-sort graphs

A Hamiltonian cycle C=⟨ u 1, u 2, …, u n(G), u 1 ⟩ with n(G)=number of vertices of G, is a cycle C(u 1; G), where u 1 is the beginning and ending vertex and u i is the ith vertex in C and u i ≠u j for any i≠j, 1≤i, j≤n(G). A set of Hamiltonian cycles {C 1, C 2, …, C k } of G is mutually independent if any two different Hamiltonian cycles are independent. For a hamiltonian graph G, the mutually independent Hamiltonianicity number of G, denoted by h(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. In this paper, we prove that h(B n )=n−1 if n≥4, where B n is the n-dimensional bubble-sort graph.

Super line-connectivity of consecutive- d digraphs

Abstract The concept of a consecutive- d digraph was proposed by Du, Hsu and Hwang as a generalization of many digraphs, such as de Bruijn digraphs, Kautz digraphs, and Imase-Itoh digraphs, which contain many hamiltonian digraphs with near-minimum diameter and near-maximum connectivity. In this paper, we show sufficient conditions for modified consecutive- d digraphs to have super line-connectivity.

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 ⌋ .

Partitionable starters for twin prime power type

Abstract Skew starters, balanced starters, partitionable starters are used in the construction of various combinatorial designs and configurations such as Room squares, Howell designs and Howell rotations. In this paper, we construct partitionable starters of order n when n is a product of two prime powers differing by 2. These partitionable starters are shown to be skew for n ⩾ 143. The results imply the existence of certain balanced Howell rotations. Moreover, we show the existence of partionable balanced starters of order n = 2m −1.