# Prof. Dr. Shin-Shin Kao

**University:**Chung-Yuan Christian University

**Department:**Department of Applied Mathematics

**Research Fields:**Combinatorics, Graph theory in Interconnection Networks.

## Publications

##### 7 On Chvátal’s Conjecture for the Hamiltonicity of 1-Tough Graphs and Their Complements

**Authors:**
Shin-Shin Kao,
Yuan-Kang Shih,
Hsun Su

**Abstract:**

In this paper, we show that the conjecture of Chv tal, which states that any 1-tough graph is either a Hamiltonian graph or its complement contains a specific graph denoted by F, does not hold in general. More precisely, it is true only for graphs with six or seven vertices, and is false for graphs with eight or more vertices. A theorem is derived as a correction for the conjecture.

**Keywords:**
Hamiltonian,
complement,
degree sum,
tough

##### 6 Myths and Strategies for Teaching Calculus in English for Taiwanese Students: A Report Based on Three-Years of Practice

**Authors:**
Shin-Shin Kao

**Abstract:**

**Keywords:**
English,
Calculus,
vocabulary,
teaching strategy,
teaching evaluation

##### 5 Hamiltonian Related Properties with and without Faults of the Dual-Cube Interconnection Network and Their Variations

**Authors:**
Shih-Yan Chen,
Shin-Shin Kao

**Abstract:**

**Keywords:**
dual-cubes,
dual-cube extensive networks,
hypercubes,
fault-tolerant
hamiltonian property,
dual-cube-like
networks

##### 4 A Further Study on the 4-Ordered Property of Some Chordal Ring Networks

**Authors:**
Shin-Shin Kao,
Hsiu-Chunj Pan

**Abstract:**

Given a graph G. A cycle of G is a sequence of vertices of G such that the first and the last vertices are the same. A hamiltonian cycle of G is a cycle containing all vertices of G. The graph G is k-ordered (resp. k-ordered hamiltonian) if for any sequence of k distinct vertices of G, there exists a cycle (resp. hamiltonian cycle) in G containing these k vertices in the specified order. Obviously, any cycle in a graph is 1-ordered, 2-ordered and 3- ordered. Thus the study of any graph being k-ordered (resp. k-ordered hamiltonian) always starts with k = 4. Most studies about this topic work on graphs with no real applications. To our knowledge, the chordal ring families were the first one utilized as the underlying topology in interconnection networks and shown to be 4-ordered. Furthermore, based on our computer experimental results, it was conjectured that some of them are 4-ordered hamiltonian. In this paper, we intend to give some possible directions in proving the conjecture.

**Keywords:**
Hamiltonian cycle,
Chordal rings

##### 3 An Improved Construction Method for MIHCs on Cycle Composition Networks

**Authors:**
Hsun Su,
Yuan-Kang Shih,
Shin-Shin Kao

**Abstract:**

Many well-known interconnection networks, such as kary n-cubes, recursive circulant graphs, generalized recursive circulant graphs, circulant graphs and so on, are shown to belong to the family of cycle composition networks. Recently, various studies about mutually independent hamiltonian cycles, abbreviated as MIHC-s, on interconnection networks are published. In this paper, using an improved construction method, we obtain MIHC-s on cycle composition networks with a much weaker condition than the known result. In fact, we established the existence of MIHC-s in the cycle composition networks and the result is optimal in the sense that the number of MIHC-s we constructed is maximal.

**Keywords:**
Hamiltonian cycle,
k-ary n-cube,
cycle composition networks,
mutually independent

##### 2 The Spanning Laceability of k-ary n-cubes when k is Even

**Authors:**
Yuan-Kang Shih,
Shu-Li Chang,
Shin-Shin Kao

**Abstract:**

**Keywords:**
Hamiltonian,
container,
k-ary n-cube,
m*-connected

##### 1 The Panpositionable Hamiltonicity of k-ary n-cubes

**Authors:**
Chia-Jung Tsai,
Shin-Shin Kao

**Abstract:**

**Keywords:**
Hamiltonian,
k-ary n-cube,
panpositionable,
bipanpositionable

## Abstracts

##### 4 On Chvátal’s Conjecture for the Hamiltonicity of 1-Tough Graphs and Their Complements

**Authors:**
Shin-Shin Kao,
Yuan-Kang Shih,
Hsun Su

**Abstract:**

**Keywords:**
Hamiltonian,
complement,
degree sum,
tough

##### 3 Myths and Strategies for Teaching Calculus in English for Taiwanese Students: A Report Based on Three-Years of Practice

**Authors:**
Shin-Shin Kao

**Abstract:**

**Keywords:**
English,
Calculus,
vocabulary,
teaching strategy,
teaching evaluation

##### 2 Hamiltonian Related Properties with and without Faults of the Dual-Cube Interconnection Network and Their Variations

**Authors:**
Shih-Yan Chen,
Shin-Shin Kao

**Abstract:**

**Keywords:**
dual-cubes,
dual-cube extensive networks,
dual-cube-like networks,
hypercubes,
fault-tolerant hamiltonian property

##### 1 A Further Study on the 4-Ordered Property of Some Chordal Ring Networks

**Authors:**
Shin-Shin Kao,
Hsiu-Chunj Pan

**Abstract:**

**Keywords:**
Hamiltonian cycle,
Chordal rings