**2014 組合數學新苗研討會**

### ⼤學數學

### 2014 8 2 3

### ⼤會演講

### 李國偉 教授 中央研究院

### 林強 教授 中央大學數學系

### 台灣師範大學公館校區 科教大樓 5 樓演講廳

### 科技部自然司數學研究推動中心

### 國家理論科學研究中心數學組 (臺北辦公室) 東賢實業股份有限公司

### 游森棚 (台灣師範大學數學系)

### 林延輯 (台灣師範大學數學系)

### 郭君逸 (台灣師範大學數學系)

### 徐泰煒 (台灣師範大學僑先部)

### 張飛黃 (台灣師範大學僑先部)

### ⼤會宗旨

組合數學新苗研討會最初由中央研究院數學研究所李國偉教授在 1990 年發起，

每年由各大學或相關研究機構輪流主辦，此研討會所舉辦的目的在於提供台灣組 合數學相關領域的應屆碩博畢業生與相關研究人員，發表其研究成果，互相交流 或討論，以獲得後續研究方向的機會。除此之外，也提供與會者互相聯絡、切磋及 觀摩的平台。

適逢今年是第 25 週年舉辦，我們邀請李國偉教授，與林強教授，擔任大會演 講，兩位資深教授在組合領域上有相當豐碩的研究成果，同時，他們也都是第一 屆新苗研討會的發起者，相信其研究經驗能夠給予後輩良好的啟發與鼓勵。此研 討會對於鼓勵青年新進之輩與國內學者專家之研究交流有很大的助益，故組合數 學新苗研討會，一直是台灣組合數學界中相當重要的研討會之一。此外，於今年 還特地邀請大陸與日本組合領域的博士生一起來參與盛會，相信藉由這樣的交流，

能為組合新苗研討會帶來新的風貌。

i

ii

### 新苗

新苗研討會是台灣組合數學界的年度盛事。只要在國內的組合數學圈子，提起

新苗一定有很多故事可以說 — 我還記得某一次茶敘時大家笑著說：「“離散”與

“組合”這兩個字面意義相反的詞居然是同一件事!」

開辦新苗研討會是令人佩服的遠見。多年來，新苗見證了數以百計的新科碩士 與博士的畢業，培養了許多人才，見證了台灣組合數學從無到有的興盛。組合數學 這個相對年輕的領域能夠在台灣形成一個不可忽視的研究聚落，新苗研討會開疆 闢土的開拓功不可沒。

而新苗今年滿二十五歲了 — 和應屆的碩士畢業生一樣大！一個研討會可以二 十五週年真是不簡單的。籌備委員會花了許多的時間，上下蒐羅迄今的所有新苗 書面議程，放在今年的網頁上。非常幸運的是，我們找到了第一屆的議程。1990 年的第一屆新苗在中央研究院舉行，六個 sessions 的主持人中，包括了發起人李國 偉教授與今年即將榮退的林強教授。多年來兩位教授在台灣組合界培育了許多學 者持續活躍於學界，對台灣組合界有很大的貢獻。我們非常榮幸邀請兩位教授擔 任今年的大會演講。

承襲往年作法，今年我們亦有最佳論文的評選。但無論有沒有得獎，都祝福所 有從新苗畢業的新科碩士與博士。各位是台灣組合數學界的新苗，有朝一日將成 為參天大樹。恭喜畢業!

謹代表籌備委員會歡迎大家蒞臨今年的新苗研討會。這是台師大首度承辦，雖 只有短短的兩天，我們仍希望能盡力做到最好。張飛黃與郭君逸兩位教授扛下了 大部分的籌備工作，林延輯與徐泰煒 (第一屆新苗的演講者，亦為林強教授的第一 位學生) 兩位教授亦協助甚多。倘有不盡完美之處，就請多多包涵。

二十五年是一個里程碑。承先啟後，讓我們一起努力，往下一個二十五年邁進。

國立台灣師範大學數學系

“教官”/游森棚 2014/8/2

iii

**Contents**

⼤會宗旨 **i**

新苗 **iii**

錄 **iv**

議程 **vi**

⼤會演講 **1**

⼤學數學 . . . 1

研 . . . 2

邀請演講 **3**
⼤學數學 . . . 3

⼤ . . . 4

⼤ . . . 4

⼤ ⼤ 研 . . . 5

博⼠ **6**
⼤學數學 . . . 6

⼤學 數 . . . 7

⼤學 數 . . . 8

⼤學 數 . . . 10

⼤學 數 . . . 11

⼤學數學 . . . 11

⼤學 數 . . . 12

⼤學 數 . . . 13

⼤學數學 . . . 14

iv

碩⼠ **15**

⼤學數學 . . . 15

⼤學 數 . . . 15

⼤學數學 . . . 16

⼤學 數 . . . 16

⼤學 數 . . . 17

⼤學 程學 . . . 17

博 ⼤學 數 . . . 18

⼤學 數 . . . 19

⼤學 數 . . . 19

⼤學數學 . . . 20

⼤學 數 . . . 21

⼤學 數 . . . 21

⼤學 數 . . . 22

⼤學數學 . . . 22

⼤學 數 . . . 23

附錄 **25**
. . . 25

會 . . . 26

新苗 . . . 29

Memo . . . 30

v

**2014 組合數學新苗研討會議程**

8 2 ( )

08:45–09:30 ( ⼤ )

09:30–09:45

09:50–10:35 ⼤會演講 ( 數學)

Graph labellings: splittable graph and antimagic graphs

10:35–11:00 ( 數學)

11:00–11:20

Session 1 501 503

11:20–11:40 ( 數學) ( 數學)

11:40–12:00 ( ⼤ 數) ( ⼤ 數)

12:00–14:00 14:00–14:25

邀請演講 ( ⼤ )

The spectral redius of connected graphs with the independence number

14:25–14:50 邀請演講 ( ⼤ )

On aﬃne-invariant strictly cyclic Steiner quadruple systems 14:50–15:10

Session 2 501 503

15:10–15:30 ( ⼤ 數) ( ⼤ 數)

15:30–15:50 ( ⼤ ) ( 數)

15:50–16:10 博 ( ⼤ 數) ( 數學)

Session 3 501 503

16:20–16:40 ( ⼤ 數) ( ⼤數學)

16:40–17:00 ( ⼤ 數) ( ⼤ 數)

17:00–17:20 ( ⼤ 數)

17:20– ⼤合 &

vi

8 3 ( )

08:30–09:15 ⼤會演講 ( 研 數學 )

Rota’s Lessons 09:15–09:40

邀請演講 ( 數學)

Lovász Local Lemma, Entropy Compression and Graph Colorings

09:40–10:00 Session 4

10:00–10:25 ( 研 數學 )

10:25–10:50 ( ⼤ 數)

10:50–11:15 ( ⼤ 數)

Session 5

11:25–11:50 ( 數學)

11:50–12:15 ( 數學)

12:15–14:00 14:00–14:25

邀請演講 ( ⼤ )

Which connected graphs are determined by their distance spectra

14:25–14:50 ( ⼤ 數)

Session 6

15:00–15:25 ( ⼤ 數)

15:25–15:50 ( ⼤數學)

16:00–16:30

vii

### ⼤會演講

### Graph Labellings: Splittable Graphs and Antimagic Graphs

### Chiang Lin ( )

Department of Mathematics, National Central University ( ⼤學數學 ) 1. Splittable Graph.

*Let G be a graph. Then G is t-splittable if the edges of G can be partitioned*
*into t isomorphic graphs. A trivial necessary condition for G to be t-splittable*
is that*|E(G)| = 0 (mod t).*

*(1) We give the necessary and suﬃcient conditions for the spiders to be t-*
splittable.

(2) We investigate the 2-splittabilities of multipaths and multicycles with mul- ticity 2, respectively.

2. Antimagic Graphs.

*Let G be a graph. If f : E(G)* *→ A where A ⊂ C, then for v ∈ V (G), the*
*vertex sum of f at v is f*^{+}*(v) =* ∑

*uv**∈E(G)**f (uv). If there exists f : E(G)* *→*
*{1, 2, . . . , |E(G)|}, 1-1 such that all vertex sums of f are distinct, then G is an-*
*timagic. A well-known conjecture is that every simple connected graph except*
*K*_{2} is antimagic. We investigate the antimagicnesses of disconnected graphs
and multigraphs.

(a) We investigate the antimagicness of star forests.

(b) The union of a path (of order *≥ 3) and a star (of order ≥ 3) is antimagic.*

(c) Any multipath with multiplicity 2 and maximum degree*≤ 3 is antimagic.*

3. A Generalization of Antimagic Graphs.

*Let G be a graph. If A⊂ C with |A| = |E(G)| and there exists f : E(G) → A,*
*1-1 such that all f*^{+}*(x) (x* *∈ V (G)) are distinct, then G is A-antimagic. If*
*B* *⊂ C with |B| ≥ |E(G)|, and G is A-antimagic for every A with A ⊂ B and*

*|A| = |E(G)|, then G is B-antimagic. Trivially, for a graph G, if B*1 *⊂ B*2 *⊂ C,*

*|B*1*| ≥ |E(G)|, and G is B*2*-antimagic, then G is B*_{1}-antimagic.

*(a) Let G be a graph. Then G isR-antimagic if and only if G is C-antimagic.*

(b) Any path of order *≥ 5 is R-antimagic.*

### Rota’s Lessons

### Ko-Wei Lih ( )

Institute of Mathematics, Academia Sinica ( 研 數學 )

Gian-Carlo Rota(1932 – 1999) 20 組合數學 數學

學 數學 學合 Rota 學 博

Fabrizio Palombi “Indiscrete Thoughts”

演講 18 “Ten Lessons I Wish I Had Been Taught” 19 “Ten Lessons for the Survival of a Mathematics Department”

Rota

2

### 邀請演講

### Lovász Local Lemma, Entropy Compression and Graph Colorings

### Hsin-Hao Lai ( )

Department of Mathematics, National Kaohsiung Normal University

( ⼤學數學 )

In the 70’s, Lovász introduced the Lovász local lemma. Lovász local lemma is a powerful probabilistic method and has been used widely in the study of graph colorings.

In [1], an inspiring method, called the entropy compression, was introduced. It is an algorithmic version of Lovász local lemma and has been used in the study of graph colorings.

In this talk, I will introduce the relation between Lovász local lemma and entropy compression. Also, I will introduce the results of graph colorings obtained by entropy compression and describe the idea behind the method.

**Keywords: Lovász Local Lemma, Entropy Compression, Graph Coloring.**

**Reference**

[1] R. Moser, G. Tardos, A constructive proof of the general Lovász local lemma, J.

ACM 57 (2010), Art. 11.

3

### Which connected graphs are determined by their distance spectra

### Huiqiu Lin ( )

Department of Mathematics, East China University of

Science and Technology ( ⼤ )

*The distance matrix D(G) = (d** _{ij}*)

_{n}

_{×n}*of a connected graph G is the matrix*

*indexed by the vertices of G, where d*

_{ij}*denotes the distance between the vertices v*

_{i}*and v*

_{j}*. Two nonisomorphic graphs with the same D-spectra are called cospectral.*

*We say that a graph is determined by the D-spectra if there is no other nonisomorphic*
*graph with the same D-spectra. In this talk, we characterize some graphs which are*
*determined by their D-spectra.*

### The spectral radius of connected graphs with the independence number ^{*}

### Ya-Lei Jin ( )

Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University ( ⼤ )

In this talk, we investigate some properties of the Perron vector of connected
graphs. These results are used to characterize all extremal connected graphs which
attain the minimum value among the spectral radii of all connected graphs with
*order n = kα and the independence number α. Moreover, all extremal graphs which*
*attain the maximum value among the spectral radii of clique trees with order n = kα*
*and the independence number α are characterized.*

*This work is joint with my advisor Xiao-Dong Zhang( ).

4

### On aﬃne-invariant strictly cyclic Steiner quadruple systems

### Xiao-Nan Lu ( )

Graduate School of Information Science, Nagoya University, Nagoya, Japan (

⼤ ⼤ 研 )

Advisor: Professor Masakazu Jimbo

*A Steiner quadruple system of order v, denoted by SQS(v), is a pair (V,B), where*
*V is a ﬁnite set of v elements, andB is a collection of 4-elements subsets of V , called*
*blocks or quadruples, such that each 3-elements subset (triple) of V is contained in*
exactly one block in*B.*

*Let G be a permutation group on V . If G leaves* *B invariant, then G is called*
*an automorphism group of SQS(v). In particular, an SQS(v) is said to be cyclic if it*
*admits a cyclic group of order v. Accordingli,B can be partitioned into orbits under*
*the cyclic permutations. Furthermore, the orbit whose cardinality is equal to v is*
*said to be full. If all the orbits are full, then the SQS(v) is said to be strictly cyclic,*
*denoted by sSQS(v).*

*Without loss of generality, for a cyclic SQS (V,B), we can identify V with Z**v*, the
*additive group of integers modulo v. Moreover, we consider the multiplicative group*
of all units of Z*v*, say Z^{×}*v*. If an SQS admits all the elements in Z^{×}*v* as multipliers,
then it is said to be aﬃne-invariant.

*It is easy to show that an sSQS(v) exists only if v* *≡ 2, 10 (mod 24). The con-*
structions of sSQS’s was ﬁrst studied by Köhler in 1979. Köhler proposed a graph
*named after him, and proved that an sSQS(v) exists if and only if the corresponding*
*Köhler’s graph has a 1-factors. But there are few constructions known for sSQS(v)*
which are independent from Köhler’s.

*In this talk, we suppose v = 2p, where p* *≡ 1, 5 (mod 12) is an odd prime,*
*and focus on the constructions of aﬃne-invariant sSQS(v). By means of a system*
*of generators of projective special linear group PSL(2, p), we deﬁne two families of*
graphs algebraically whose 1-factors play an important role in our constructions. By
*computer search, we also verify that all these graphs for p < 100, 000 have 1-factors.*

*Futhermore, we give recursive constructions for aﬃne-invariant sSQS(2p** ^{m}*), for any

*positive integer m.*

**Keywords: Steiner quadruple system, projective linear group, 1-factor.**

5

### 博⼠

### Maximum packings and minimum coverings of multigraphs with paths and stars

### Chun-Cheng Chen ( )

Department of Mathematics, National Central University ( ⼤學數學 ) Joint work with Hung-Chih Lee( )

*Let F , G, and H be multigraphs. An (F, G)-decomposition of H is an edge*
*decomposition of H into copies of F and G using at least one of each. For subgraphs*
*L and R of H, an (F, G)-packing (resp. (F, G)-covering) of H with leave L (resp.*

*padding R) is an (F, G)-decomposition of H* *− E(L) (resp. H+E(R)). An (F, G)-*
*packing (resp. (F, G)-covering) of H with the largest (resp. smallest) cardinality*
*is a maximum (F, G)-packing (resp. minimum (F, G)-covering), and its cardinality*
*is referred to as the (F, G)-packing number (resp. (F, G)-covering number) of H.*

*Let k be a positive integer. A k-path, denoted by P*_{k}*, is a path on k vertices. A*
*k-star, denoted by S*_{k}*, is a star with k edges. In this paper, we determine the packing*
*numbers and the covering numbers of both λK**n* *and λK**n,n* *with (k + 1)-paths and*
*k-stars for any λ, n and k. Moreover, necessary and suﬃcient conditions for the*
*existence of (P*_{k+1}*, S*_{k}*)-decompositions of both λK*_{n}*and λK** _{n,n}* are given.

**Keywords: Packing, Covering, Path, Star.**

6

### Factorizations of Suﬃxes of Two-Way Inﬁnite Characteristic Words

### Fang-Yi Liao ( )

Department of Applied Mathematics,

Chung Yuan Christian University ( ⼤學 數 ) Advisor: Wai-Fong Chuan( )

*Let α be an irrational number between 0 and 1 with continued fraction expansion*
*[0; a*_{1} *+ 1, a*_{2}*, a*_{3}*,· · · ], where a**n* *≥ 1 (n ≥ 1). Deﬁne a sequence of numbers {q**n**}**n**≥1*

*by q*_{−1}*= 1, q*_{0} *= 1, q*_{n}*= a*_{n}*q*_{n}_{−1}*+ q*_{n}_{−2}*(n* *≥ 1). For each integer k ≥ −1, we con-*
*sider the kth-order factorization of each suﬃx H of a two-way inﬁnite characteristic*
*word of α of the form: H = u*_{k}*u*_{k+1}*u*_{k+2}*· · · , where the length of the factor u**i* is
*q*_{i}*(i* *≥ k). We show that in such a factorization, either all u**i* are singular words,
*or there exists a nonnegative integer q such that H begins with q singular words,*
*and u*_{k+q}*, u*_{k+q+1}*, u*_{k+q+2}*,· · · are α-words. Moreover, the labels of these α-words are*
*uniquely determined by the F -representation of a nonnegative integer obtained from*
*the position of H in the two-way inﬁnite characteristic word.*

*Next, deﬁne Markov word pattern of order k (k* *≥ −1) generated by a pair of*
*seed words (u, v) as follows: M*_{k}*(u, v) = z*_{1}*z*_{2}*z*_{3}*· · · , where z*1 *= u, z*_{2} *= v, and*
*z*_{i}*= z*_{i}^{a}_{−1}^{i+k}^{−1}^{−1}*z*_{i}_{−2}*z*_{i}_{−1}*(i* *≥ 3). We show that each suﬃx H of each two-way inﬁnite*
characteristic word is a Markov word pattern, and the pairs of its seed words obtained
*are adjacent α-words; we also ﬁnd all possible pairs of seed words of H which are*
*pairs of adjacent α-words. On the other hand, we show that each Markov word*
*pattern generated by any pair of adjacent α-words is a suﬃx of a two-way inﬁnite*
characteristic word.

*Finally, we study the set V of all pairs of adjacent α-words. We describe the*
*elements of V in terms of cyclic shifts, lexicographic order, and labels. For each*
*α-word w, we ﬁnd all possible words u such that (u, w) (resp., (w, u)) are pairs of*
**adjacent α-words. Also, we construct a directed graph G*** _{α}* consisting of the vertex

*set of all α-words and the edge set V , and use such a graph to illustrate the results*

*about adjacent α-words, and the suﬃxes of the two-way inﬁnite characteristic words*that they generate.

**Keywords: Two-way inﬁnite characteristic word, Factorization, Markov word**
*pattern, Seed word, Adjacent α-word.*

7

### Self-stabilizing Minimal Dominating Set Algorithms of Distributed Systems and the Signed Star Domination Number

### of Cayley Graphs

### Well Y. Chiu ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Chiuyuan Chen( )

The study of the domination problem in graph theory began in the nineteen-
sixties. A distributed system such as an ad hoc network can be modeled by an
*undirected simple graph G = (V, E), where V represents the set of nodes (i.e.,*
*processes) and E represents the set of interconnections between processes of the*
*distributed system. A subset D of the vertex set V of G is a dominating set if each*
*vertex v∈ V is either a member of D or adjacent to a vertex in D. A dominating set*
*of G is a minimal dominating set (MDS) if none of its proper subsets is a dominating*
*set of G. An MDS has an application of clustering in wireless networks and is*
maintained for minimizing the number of required resource centers.

Self-stabilization is a concept of designing a distributed system for transient fault toleration and was introduced by Dijkstra in 1974. A distributed system is self- stabilizing if, regardless of its initial conﬁguration, the system is guaranteed to reach a legitimate (i.e., correct) conﬁguration in a ﬁnite time. Here the system conﬁgu- ration consists of the state of every process. A self-stabilizing algorithm comprises a collection of rules and each rule has a trigger precondition and an action. The action changes the state of the node by updating its variables. An execution of a rule is called a move. The performance of the proposed algorithms of this thesis is measured by the total number of moves executed by an algorithm. Various execution models have been used in self-stabilizing algorithms and these are encapsulated with the notion of daemons. A daemon can be fair or unfair. It is well-known that an unfair distributed daemon is more practical than other types of daemons.

*Let n denote the number of nodes (processes) in a given distributed system. In*
2007, Turau proposed the ﬁrst linear-time self-stabilizing algorithm for the MDS
problem under an unfair distributed daemon; this algorithm stabilizes in at most
*9n moves. In 2008, Goddard et al. improved the result to a 5n-move algorithm.*

It is interesting to develop an algorithm that takes less moves than the best known
*result—5n moves using an unfair distributed daemon. In this thesis, we will present*
*a 4n-move self-stabilizing MDS algorithm using an unfair distributed daemon.*

8

It is desired that an MDS algorithm is MDS-silent, which means that if the orig- inal conﬁguration of the distributed system is already an MDS, then the algorithm should not make any move. Note that in the normal model, a node can only access the information of its 1-hop neighbors and we call such information distance-1 in- formation. Unfortunately, in this thesis we will prove that distance-1 information is not suﬃcient for building up an MDS-silent algorithm for a distributed system.

*What will happen if a node can access the information of its k-hop neighbors for*
*k* *≥ 2? In this thesis, we will discuss this problem and propose a new performance*
measure, called stableness, for self-stabilizing MDS algorithms. We also generalize
this result to categorize all self-stabilizing algorithms into four levels. In particular,
we will show that a self-stabilizing MDS-silent algorithm can be built up under the
*distance-2 model and the stabilizing time is upper bounded by 2n.*

*Let G be a simple connected graph with vertex set V (G) and edge set E(G). A*
*function f : E(G)* *→ {−1, 1} is called a signed star dominating function (SSDF)*
*on G if* ∑

*e**∈E**G**(v)**f (e)* *≥ 1 for every v ∈ V (G), where E**G**(v) is the set of all edges*
*incident to v. The signed star domination number of G is deﬁned as γ**SS**(G) =*
min*{*∑

*e**∈E(G)**f (e)| f is an SSDF on G}. Let D be a ﬁnite digraph with vertex*
*set V (D) and arc set A(D). For each vertex v* *∈ V (D), let A(v) be the set of all*
*out-going arcs from v. By replacing E(v) by A(v), one can deﬁne SSDF on D and*
*γ*_{SS}*(D) = min{*∑

*a**∈A(D)**f (a)* *| f is an SSDF on D}. Let Γ be a ﬁnite nontrivial*
*group and S be a nonempty subset of Γ. The Cayley digraph Cay*_{D}*(Γ, S) is the*
*digraph whose vertices are the elements of Γ, and there is an arc from α to ασ*
*whenever α∈ Γ and σ ∈ S. Let Ω be a symmetric generating subset of nonidentity*
*elements of Γ. The Cayley graph Cay(Γ, Ω) corresponding to Γ and Ω is the ordinary*
*graph with vertex set Γ and edge set E ={{α, ασ} | α ∈ Γ, σ ∈ Ω}. In this thesis,*
we obtain exact values for the signed star domination number of all Cayley digraphs
*Cay*_{D}*(Γ, S) and certain classes of Cayley graphs Cay(Γ, Ω), which is later generalized*
to*{2, 1}-factorable graphs.*

**Keywords: self-stabilizing, minimal dominating set, stableness, signed star dom-**
ination number.

9

### Decycling Number on Graphs and Digraphs

### Min-Yun Lien ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Hung-Lin Fu( )

A set of vertices of a graph or an digraph whose removal induces an acyclic graph
*is referred as a decycling set, or a feedback vertex set, of the graph. The minimum*
*cardinality of a decycling set of a graph G is referred to as the decycling number of*
*G.*

*The problem of determining the decycling number has been proved to be NP-*
complete for general graphs, which also shows that even for planar graphs, bipartite
graphs and perfect graphs, the computation complexity of ﬁnding their decycling
numbers is not reduced.

The problem of destroying all cycles in a graph by deleting a set of vertices originated from applications in combinatorial circuit design. Also, it has found ap- plications in deadlock prevention in operating systems, the constraint satisfaction problem and Bayesian inference in artiﬁcial intelligence, monopolies in synchronous distributed systems, the converters’ placement problem in optical networks, and VLSI chip design.

In this talk, we introduce the decycling number of graphs and also digraphs.

*The graphs we consider are outerplanar graphs and grid graphs P*_{m}*P**n*. For the
ﬁrst class of graphs, we characterize their decycling number by way of the cycle
packing number and for grid graphs, we improve the known results to obtain either
tight bounds or exact values. On digraphs, we consider generalized Kautz digraphs
and generalized de Bruijn digraphs. Mainly, we use a novel idea in which we ﬁnd
a sequence of subsets of vertex set satisfying certain conditions and then obtain a
decycling set. This provides an upper bound of the decycling number of digraphs
we consider. Note that this idea can be applied to ﬁnd the decycling set of general
digraphs.

**Keywords: Decycling Number, Feedback vertex number, Outerplanar Graphs,**
*P**m**P**n*, Generalized Kautz Digraphs, Generalized de Bruijn Digraphs.

10

*Optimally ℓ-Pebbling Cycles*

### H. X. Chiang ( )

Department of Mathematics,

Chung Yuan Christian University ( ⼤學 數 ) Joint work with Chin-Lin Shiue and M. M. Wong

*Let G be a graph. A distribution of pebbles of G is a a function δ : V (G)* *→*
*N* *∪ {0}. In a graph with a distribution of pebbles, a pebbling move consists of*
removing two pebbles from one vertex and then placing one pebble at an adjacent
*vertex. A pebbling type α of G is a mapping from V (G) into*N∪

*{0}. A distribution*
*δ is called an α-pebbling if whenever we choose a target vertex v, we can move*
*α(v) pebbles to v by applying pebbling moves repeatedly ( if necessary). An it ℓ-*
*pebbling of G is an α-pebbling of G for which α(v) = ℓ for each v* *∈ V (G), where*
*ℓ is a positive integer. The optimal α-pebbling number f*_{α}^{′}*(G) of G is the minimum*
*number of pebbles used in an α-pebbling of G. We denote f*_{α}^{′}*(G) = f*_{ℓ}^{′}*(G) if α(v) = ℓ*
*for each v* *∈ V (G). The optimal pebbling number of a graph G is the optimal 1-*
*pebbling number of G and denoted by f*^{′}*(G). In this paper, we ﬁrst ﬁnd the optimal*
*2-pebbling numbers and optimal 3-pebbling numbers of the cycle graph C** _{n}*. Second,

*we ﬁnd the upper bound and lower bound of C*

*n*. Lastly apply them to aproach

*optimal ℓ-Pebbling of C*

*.*

_{n}**Keywords: optimal pebbling number, cycle.**

### A study of bull-design

### Shu-Wen Lo ( )

Department of Mathematics, Tamkang University ( ⼤學數學 ) Advisor: Chin-Mei Kau Fu( )

*A complete graph K*_{n}*is a simple graph of order n whose vertices are pairwise*
*adjacent. A decomposition of graph G is a collection* *H = {H*1*, H*_{2}*, . . . , H*_{k}*} of*
*subgraphs of G, such that E(H** _{i}*)

*∩ E(H*

*j*) =

*∅ (i ̸= j) and E(H*1)

*∪ E(H*2)

*∪ · · · ∪*

*E(H*

_{k}*) = E(G). Let H be a graph, an H-design of a complete graph K*

*, denoted by*

_{n}*(K*

_{n}*, H), is a pair (X,B), where X is the vertex set of the complete graph K*

*n*and

*B*

11

*is a collection of subgraphs of K** _{n}*, called blocks, such that each block is isomorphic

*to H, and any edge of K*

_{n}*is contained in exactly one subgraph of K*

*. Furthermore,*

_{n}*G has an H-decomposition or G can be decomposed into H, if H*

_{i}*is isomorphic to H*(1

*≤ i ≤ k). Therefore, a (K*

*n*

*, H)-design exists means K*

_{n}*has an H-decomposition.*

*A bull is a graph B which is obtained by attaching two edges to two vertices of*
*a triangle. A (K*_{n}*,B)-design is called a bull-design of order n.*

In Chapter 2, we show that the necessary and suﬃcient condition of a bull-design
*of order n exist precisely when n≡ 0, 1 (mod 5).*

*In Chapter 3, we consider the maximum packing of bull-design of order n. We*
*obtain that the leave of maximum packing is a set of one edge if n≡ 2 or 4 (mod 5)*
*and a set of three edges if n* *≡ 3 (mod 5). By the above results, we obtain that*
the necessary and suﬃcient conditions for the existence of bull-designs of a complete
*multi-partite graph λK*_{n}*are the follows: λ* *̸≡ 0 (mod 5) and n ≡ 0, 1 (mod 5), or*
*λ≡ 0 (mod 5) and for all n.*

*In Chapter 4, we obtain that the spectrum of bull-design of order n intersecting*
*in pairwise disjoint blocks is 0, 1, 2, . . . , [n/5], when n > 5 and n≡ 0, 1 (mod 5), and*
the spectrum of bull-design of order 5 intersecting in pairwise disjoint blocks is 0.

We also show that the spectrum of triangle intersection numbers of two bull-design
*of order n is 0, 1, 2, . . . , n(n− 1)/10, for n ≡ 0, 1 (mod 5).*

*In Chapter 5, we obtain that a bull-design of order n can be embedded in a*
*bull-design of order m if and only if m* *≥ 3n/2 + 1 or m = n. This produces a*
generalization of the Doyen-Wilson theorem for bull-designs.

**Keywords: Decomposition, bull-design, Packing, Intersection, Doyen-Wilson**
theorem.

### A General Framework for Central Limit Theorems of Additive Shape Parameters in Random Digital

### Trees

### Chung-Kuei Lee ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 )

Tries, PATRICIA tries and (bucket) Digital Search Trees are fundamental data structures in computer science with numerous applications. In recent papers, a

12

general framework for obtaining the mean and variance of additive shape parameters of tries, PATRICIA tries and DSTs under the Bernoulli model was proposed. Later on, we showed that a slight modiﬁcation of the frameworks yields a central limit theorem for shape parameters, too. This central limit theorem contains many of the previous central limit theorems from the literature and it can be used to prove recent conjectures and derive new results. As an example of the trie case, we will consider a reﬁnement of the size of tries and PATRICIA tries, namely, the number of nodes of ﬁxed outdegree and obtain (univariate and bivariate) central limit theorems. For the DSTs case, we use 2-protected node as an example to illustrate how the framework will work.

### Spectral Excess Theorem and its Applications

### Guang-Siang Lee ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Chih-wen Weng( )

The spectral excess theorem gives a quasi-spectral characterization for a regular
graph to be distance-regular. An example demonstrates that this theorem cannot
directly apply to nonregular graphs. In order to make it applicable to nonregular
graphs, a ‘weighted’ version of the spectral excess theorem is given. For application,
*we show that a graph with odd-girth 2d + 1 is distance-regular, generalizing a result*
of van Dam and Haemers. We then apply this line of study to the class of bipartite
graphs. It is well-known that the halved graphs of a bipartite distance-regular graph
are distance-regular. Examples are given to show that the converse does not hold.

Thus, a natural question is to ﬁnd out when the converse is true. We give a quasi- spectral characterization of a connected bipartite weighted 2-punctually distance- regular graph whose halved graphs are distance-regular. In the case the spectral diameter is even we show that the graph characterized above is distance-regular.

**Keywords: Distance-regular graph, Distance matrices, Predistance polynomials,**
Spectral diameter, Spectral excess theorem.

13

### Covering Problems in Graphs

### Sheng-Hua Chen ( )

Department of Mathematics, National Taiwan University ( ⼤學數學 ) Advisor: Gerard Jennhwa Chang( )

Covering problems in graphs are optimization problems about covering the vertex
*set V (G) or the edge set E(G) of a graph G under some additional restrictions. In*
*other words, a graph covering of G is a collection of vertex/edge subsets of G such*
*that each vertex or each edge of G is belonged to at least one subset in this collection.*

Graph covering enjoys many practical applications as well as theoretical challenges. It is heavily used in various ﬁelds such as biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling) etc.

In this thesis, we study six covering problems in graphs, which study the optimally
*of the following related functions. A strong edge-coloring is a function that assigns*
to each edge a color such that any two edges within distance two apart receive
*diﬀerent colors. An edge Roman dominating function is the edge version of a Roman*
*dominating function, that is, a function f : E(G)→ {0, 1, 2} such that every edge e*
*with f (e) = 0 is adjacent to some edge e*^{′}*with f (e** ^{′}*) = 2. More generally, for a ﬁxed

*positive integer k, a k-power Roman dominating function is a function f : V (G)→*

*{0, 1, . . . , k} such that every vertex u with f(u) = 0 is adjacent to some vertex v*

*with f (v) = i≤ k within distance i. A distance edge cover is a generalization of an*

*edge cover, that assigns each edge a label such that every vertex is within distance j*

*from some edge with label j. A modular orientation is an orientation of edges such*

*that the in-degree equals to the out-degree of each vertex. A relaxation procedure*is a series of relabeling by changing the sign of a negative vertex and sharing the diﬀerence equally to its neighbors.

The purpose of this thesis is to study the above mentioned graph covering prob- lems from algorithmic, algebraic and probabilistic point of view. In particular, we give exact values and/or upper/lower bounds for related parameters of these prob- lems. New technics are developed to established interesting results, including the proofs/dis-proofs of some known conjectures.

14

### 碩⼠

### The 3-split of multipaths and multicycles with multiplicity 2

### Meng-Ping Hsieh ( )

Department of Mathematics,

National Central University ( ⼤學數學 ) Advisor: Chiang Lin( )

*Let G be a graph and t be a positive integer. A t-split of G is a partition of the*
*edges of G into t isomorphic subgraphs. A graph is said to be t-splittable if it has a*
*t-split.*

In this thesis we prove the following results.

*Theorem. Let Q be a multipath with multiplicity 2 such that* *|E(Q)|=0 (mod 3).*

*Then Q is 3-splittable.*

*Theorem. Let C be a multicycle with multiplicity 2 such that* *|E(C)|=0 (mod 3).*

*Then C is 3-splittable.*

### Diﬀy Hexagons

### Wei-Ming Wang ( )

Department of Mathematical Science, National Chengchi University ( ⼤學 數 )

Advisor: Young-Ming Chen( )

In this thesis, we study the Diﬀy Hexagons: Initially, we regard a Ducci se- quence as a Diﬀy Hexagon game and discuss some properties about Ducci sequences.

However, a Ducci sequence isn’t actually a Diﬀy Hexagon game due to the fact that regular hexagons has some symmetries under rotations and reﬂections, but the Ducci sequences don’t. Therefore, we apply an identiﬁcation in the end.

**Keywords: Cycles, Diﬀy Hexagons, Ducci processes, Ducci sequences, Periods,**
Similar cycles.

15

### On the distribution of the leading statistic for the bounded deviated permutations

### Wei-Liang Chien ( )

Department of Mathematics,

National Taiwan Normal University ( ⼤學數學 ) Advisor: Yen-Chi R. Lin( )

The purpose of the thesis is to investigate the distribution of the initial number
*in the bounded deviated permutations S*_{n+1}* ^{ℓ,r}* , assuming the uniform distribution in

*S*

_{n+1}*.*

^{ℓ,r}*Deﬁne the random variable X*_{n}*to take the value k if π*_{1} *= k+1 π = π*_{1}*π*_{2}*· · · π**n+1**∈*
*S*_{n+1}^{ℓ,r}*. By considering the bivariate generating function A(z, u), we could calculate the*
*expected value and the standard deviation for X** _{n}*. The method is then applied to two

*speciﬁc cases, S*

_{n+1}

^{1,2}*and S*

_{n+1}

^{2,2}*. Since the coeﬃcients λ*

*of the bivariate generating function do not have a closed form, we will apply the Hayman method to get its asymptotic formula. Finally, by running computer programs, the convegences of the*

_{n,k}*normal distribution of S*

_{n+1}

^{1,2}*and S*

_{n+1}*are veriﬁed.*

^{2,2}### The Number of 2-Protected Nodes in Tries and PATRICIA Tries

### Guan-Ru Yu ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Michael Fuchs( )

Digital trees are data structures which are of fundamental importance in Com- puter Science. Recently, so-called 2-protected nodes have attracted a lot of attention.

For instance, J. Gaither, Y. Homma, M. Sellke, and M. D. Ward derived an asymp- totic expansion for the mean of the number of 2-protected nodes in random tries.

16

Moreover, J. Gaither and M. D. Ward considered the variance and conjectured a central limit theorem.

In this talk, we will explain our recent results on the number of 2-protected nodes in tries and PATRICIA tries. More precisely, we will derive asymptotic expansions of moments and prove the conjectured central limit theorem of J. Gaither and M.

D. Ward. An interesting aspect of our work is that our results contain divergent series which however become convergent (and yield correct results) by appealing to the theory of Abel summability.

**Keywords: digital tree, 2-protected nodes, moments, central limit theorem.**

### Perfect Secret Sharing Schemes for Access Structures Based on Graphs

### Bo-Rong Lin ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Hung-Lin Fu( ) and Hui-Chuan Lu( )

*A perfect secret sharing scheme based on a graph G is a randomized distribution*
of a secret among the vertices of the graph so that the secret can be recovered from
the information assigned to the endvertices of any edge, while the total information
assigned to an independent set of vertices is independent (in statistical sense) of the
secret itself.

*The (worst case) information ratio of G is the largest lower bound on the amount*
of information some vertex must remember for each bit of the secret. Using entropy
method, we calculate a lower bound on the information ratio for an inﬁnite class
of graphs we consider in this thesis. We also use the generalized vector space con-
struction to construct perfect secret sharing schemes with information ratio 2 for two
subclasses of graphs. This upper bounded is very close to our lower bound in some
circumstances, which means the secret sharing schemes we construct are in fact very
good.

17

### A Study on Quality-Enhanced Visual Multi-Secret Images Sharing Schemes

### Lu-Chung Chen ( )

Department of Computer Science and Information Engineering, National Chi Nan University ( ⼤學 程學 )

Advisor: Justie Su-Tzu Juan ( )

Visual secret sharing is a method of encrypting images. In 2012, a multiple
visual secret sharing scheme by shifting random grid was proposed by Chang. In this
scheme, multiple secret-images can be encrypted into two share images. However,
*the distortion of this scheme is quite high (especially when the number of secret*
images are more). In order to decrease the distortion of this scheme, we propose
three improved multiple visual secret sharing schemes by shifting random grid.

*The ﬁrst method of our schemes is increasing the number of fragments (INF, for*
*short). The second is increasing the number of shares (INS, for short). The third is*
*distributing the black pixels evenly on shares (DBPEOS, for short), this method not*
decreases the quantity of distortion but increases the evenness of black pixels.

In addition to less distortion, INS develops two new modes for mobile devices.

Therefore, our proposed schemes can be used for more applications.

**Keywords: visual secret sharing, random grid, multiple secret-images, mobile**
devices.

### The Minimum Rank of Buds

### Po-Yu Hsu ( 博 )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Chih-Wen Weng( )

*For a simple graph G of order n with vertex set [n] =* *{1, 2, · · · , n}, an n × n*
*real symmetric matrix A, whose ij-th entry is not zero if and only if there is an edge*
*joined i and j in G, is said to be associated with G. The minimum rank of G is*
deﬁned to be the smallest possible rank over all symmetric real matrices associated

18

*with G. A bud based on [n− m] is a graph G with vertex set V (G) = [n] satisfying*
the following axioms:

*(i) The subgraph of G induced on [n* *− m] is a cycle C**n**−m*, and the subgraph
*induced on [n]\ [n − m] has no edge.*

*(ii) The cycle C*_{n}_{−m}*can be parted into m disjoints paths, and the length of these*
*paths are at least 2. For all vertex v in [n]\ [n − m], v has at least three*
*neighbors in the same path. Any two vertices in [n]\ [n − m] are not connected*
to the same path.

*In the thesis we will show that a bud based on [n− m] has minimum rank n − m − 2.*

**Keywords: Graph, Minimum rank, Bud.**

### Mathematical Properties and Construction of Quantum Error Correcting Codes

### Jui-Yi, Tsai ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Chih-Wen Weng( )

This thesis introduces about the quantum error correcting codes in the viewpoint of classical error correcting codes. We also introduce some construction and charac- terization of quantum error correcting codes. Thereafter, we give a construction of quantum error correcting codes associated with graphs, which generalizes a previous result that excludes the binary case so that it is valid for all cases.

**Keywords: quantum error correcting codes,graph.**

*The study of r-locating-dominating codes in paths*

### Hsiao-Yun Lin ( )

19

Department of Applied Mathematics,

National Sun Yat-sen University ( ⼤學 數 ) Advisor: Li-Da Tong( ⼤)

The locating-dominating code of graph was introduced by Colbourn, Slater, and
*Stewart. Slater proved that M*_{1}^{LD}*(P**n*) = *⌈*^{2n}_{5} *⌉. Honkaia proved that M*2^{LD}*(P**n*) =

*⌈*^{n+1}_{3} *⌉. Exoo, Junnila, and Laihonen determined M**r*^{LD}*(P** _{n}*) for (3

*≤ r ≤ 4) and*

*(r*

*≥ 5 and 2 ≤ n ≤ 7r + 3). In this thesis, we determine M*

_{r}

^{LD}*(P*

_{n}*) for r*

*≥ 5 and*

*7r + 4≤ n ≤ 11r + 5.*

**Keywords: Locating-dominating code, Dominating, Graph.**

### The study of secure-dominating set of graph products

### Hung-Ming Chang ( )

Department of Mathematics,

National Kaohsiung Normal University ( ⼤學數學 ) Advisor: Hsin-Hao Lai( )

*If G is a graph and v is a vertex of G, then N*_{G}*(v) denotes the neighborhood of*
*v in G and N*_{G}*[v] denotes the closed neighborhood of v in G. Given a subset S of*
*V (G), a function A deﬁned on S is called an attack on S in G if A(u)⊆ N**G**(u)− S*
*for any u* *∈ S and A(u) ∩ A(v) = ∅ for any distinct vertices u, v. And a function*
*D deﬁned on S is called a defense of S if D(u)* *⊆ N**G**[u]∩ S for any u ∈ S and*
*D(u)∩ D(v) = ∅ for any distinct vertices u, v. A nonempty subset S of V (G) is*
*called a secure set of G if for each attack A on S, there exists a defense of S such*
that *|D(u)| ≥ |A(u)| for any u ∈ S.*

*One can think the vertices of A(u) as attackers of u and those of D(u) as defenders*
*of u. Each vertex not in S can attack at most one of its neighbor in S. And each*
*vertex in S can defense at most one attack on itself or one attack on its neighbors*
*in S. The attack is thwarted if* *|D(u)| ≥ |A(u)|. For a secure set S, each attack on*
*S can be thwarted.*

*A set S is a secure-dominating set of G if S is a secure set of G that is also a*
*dominating set of G. The secure-dominating number of G is the minimum cardinality*
*of secure-dominating sets of G.*

20

In this thesis, we obtain some results of secure-dominating sets and secure- dominating numbers of strong product and lexicographic product of graphs.

**Keywords: secure-dominating set, secure-dominating number, strong product,**
lexicographic product.

### The Laplacian Spectral Radius of a Graph

### Fan-Hsuan Lin ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Chih-Wen Weng( )

*Let G = (V, E) be a simple connected graph with the vertex set V and the edge*
*set E. We have a new sharp bound for the Laplacian spectral radius of G, which*
improves some known upper bounds.

**Keywords: Graph, Laplacian matrix, Laplacian spectral radius.**

### Error-correcting pooling designs and group testing for consecutive positives

### Yi-Tsz Tsai ( )

Department of Applied Mathematics,

National University of Kaohsiung ( ⼤學 數 ) Advisor: Huilan Chang( )

Pooling designs are standard experimental tools in many biotechnical applica- tions. Many famous pooling designs have been constructed from mathematical structures by “containing relation”. Recently, pooling designs constructed by “in- tersecting relation” have been proposed by Nan and Guo (2010) and Guo and Wang (2011). Constructing by intersecting relation provides much better error-tolerance capabilities. In this thesis, we study the error-tolerance capabilities of pooling de- signs constructed by intersecting relation from combinatorial structures proposed by

21

D’yachkov et al. (2007) and Bai et al. (2009). Motivated by application to DNA
sequencing, group testing for consecutive positives has been proposed by Balding
*and Torney(1997) and Colbourn (1999) where n items are linearly ordered and all*
*up to d positive items are consecutive in the order. In this thesis, we study a vari-*
*ation of (k, m, n)-selectors and use this combinatorial object to design a two-stage*
algorithm for group testing of consecutive positives. Our algorithm takes at most
12 log_{2}*⌈n/d⌉ + 14e + 3d tests to identify all positives and its decoding complexity is*
*O(*^{n}* _{d}*log

^{n}

_{d}*+ d).*

**Keywords: Group Testing, Pooling design, Error-tolerance, Consecutive Posi-**
tives.

### Threshold group testing with consecutive positives

### Yi-Lin Tsai ( )

Department of Applied Mathematics,

National University of Kaohsiung ( ⼤學 數 ) Advisor: Huilan Chang( )

Threshold group testing introduced by Damaschke (2006) is a generalization of
classical group testing where a group test yields a positive (negative) outcome if it
*contains at least u (at most l) positive items, and an arbitrary outcome for other-*
wise. Motivated by applications to DNA sequencing, group testing with consecutive
positives has been proposed by Balding and Torney (1997) and Colbourn (1999)
*where n items are linearly ordered and all up to d positive items are consecutive*
in the order. In this thesis, we use threshold-constrained group tests to deal with
group testing with consecutive positives. We prove that all positive items can be
identiﬁed in*⌈log*2(*⌈n/u⌉ − 1)⌉ + 2⌈log*2*(u + 2)⌉ + ⌈log*2*(d− u + 1)⌉ − 2 tests sequen-*
*tially for the gap-free case (u = l + 1) while the information-theoretic lower bound*
is *⌈log*2*n(d− u + 1)⌉ − 1 when n ≥ d + u − 2 and show that the case with a gap*
*(u > l + 1) can be dealt with by the subroutines used to conquer the gap-free case.*

Using a variation of cover-free family we show that the set of positives can be approx-
imately identiﬁed in 15 log_{2} ^{n}_{d}*+ 4d + 71 group tests nonadaptively and its decoding*
*complexity is O(*^{n}* _{d}*log

_{2}

^{n}

_{d}*+ ud*

^{2}).

**Keywords: Group testing, Threshold, Consecutive positives, Cover-free families.**

22

### Weight Choosability of theta Graphs

### Ting-Feng Jian ( )

Department of Mathematics,

National Taiwan University ( ⼤學數學 ) Advisor: Gerard Jennhwa Chang( )

The 1,2,3-conjecture is a problem of edge weight colorability of graphs which was
posed by M. Karoński et al in 2004. Further problem of edge weight choosability
of graphs was posed by T. Bartnicki et al in 2009. While being solved for some
special cases, the two problems are still open nowadays. In this thesis, we use the
combinatorial nullstellensatz and the permanent to ﬁnd some results. We go through
*the cycles, then discuss the θ-graphs and generalized θ-graphs. The main result of*
this thesis is to show these graphs are all 3-edge weight choosable.

**Keywords: weight choosability, 3-weight choosable, combinatorial nullstellen-**
*satz, permanent, cycles, θ-graphs, generalized θ-graphs.*

### On Antimagic Labeling and Associated Deﬁciency Problems for Graph Products

### Si-Hao Wu ( )

Department of Applied Mathematics,

National Chiao Tung University ( ⼤學 數 ) Advisor: Hung-Lin Fu( ) and Tao-Ming Wang( )

*Let G = (V (G), E(G)) be a ﬁnite simple graph with p =* *|V (G)| vertices and*
*q =* * |E(G)| edges. An antimagic labeling of G is a bijection from the set of*
edges to the set of integers

*{1, 2, · · · , q} such that the vertex sums are pairwise*distinct, where the vertex sum at a vertex is the sum of labels of all edges inci-

**dent to such vertex. Moreover G is called (a, d)-antimagic if the vertex sums are***a, a + d,· · · , a + (|V | − 1)d for some positive integers a and d. For the graph G, the*

**(a, d)-antimagic deﬁciency (antimagic deﬁciency, respectively) is deﬁned as the***minimum integer k such that the injective edge labeling f : E(G)→ {1, 2, · · · , q +k}*

23

*is (a, d)-antimagic (antimagic, respectively). This thesis mainly studies antimagic la-*
belings and associated antimagic deﬁciency problems for certain graph products. In
particular, we show the antimagic-ness for strong product of any even regular graph
*and any regular graph. Also we determine the (a, 1)-antimagic deﬁciency for the*
*Cartesian product of cycles C*_{m}_{1}*C**m*2 *and the (a, 1)-antimagic deﬁciency for the*
*strong product of cycles C*_{m}_{1} * C**m*2.

**Keywords: antimagic labeling, antimagic graph, strong product, Cartesian**
product.

24

### 附錄

### 陳聖華 (臺大數學) 羅淑玟 (淡江數學) 李忠逵 (交大應數) 連敏筠 (交大應數) 李光祥 (交大應數) 邱鈺傑 (交大應數)

### 梁育菖 (中山應數)(103.2) 林哲宇 (中山應數)(103.2) 廖芳儀 (中原應數)(103.2)

### 榮獲博士學位

### 張宏名 (高師數學) 簡維良 (台師數學) 王偉名 (政大應數) 林筱芸 (中山應數) 簡廷豐 (臺大數學) 蔡睿翊 (交大應數) 余冠儒 (交大應數) 許博喻 (交大應數) 林凡軒 (交大應數) 鄭伊婕 (交大應數) 林伯融 (交大應數) 吳熹皓 (交大應數) 謝孟萍 (中央數學) 蔡宜霖 (高大應數) 蔡一慈 (高大應數) 陳律仲 (暨大資工)

### 榮獲碩士學位

25

### 會

⼤會演講研 數學 makwlih@sinica.edu.tw

⼤學數學 lchiang@math.ncu.edu.tw

⼤學 數 tlwong@math.nsysu.edu.tw

⼤ ⼤學 數 ldtong@math.nsysu.edu.tw

⼤學數學 cyshen@math.ncu.edu.tw

⼤學數學 scliaw@math.ncu.edu.tw

⼤學 數學 clshiue@math.cycu.edu.tw

⼤學 數學 bytsai0808@gmail.com

⼤學 jungle@cute.edu.tw

⼤學 yp-tsao@gm.cute.edu.tw

⼤學 數 weitianli@dragon.nchu.edu.tw

⼤學 數 yltsai@nchu.edu.tw

mingju99@jente.edu.tw

⼤ mhhuang@mail.ypu.edu.tw

⼤學數學 gjchang@math.ntu.edu.tw

⼤學數 學 twhsu@ntnu.edu.tw

⼤學數 學 feihuang0228@gmail.com

⼤學數學 yclinpa@gmail.com

⼤學數學 junyiguo@gmail.com

⼤學數學 senpengeu@gmail.com

⼤學 數學 wuhsiunglin@nctu.edu.tw

⼤學 數學 mfuchs@math.nctu.edu.tw

⼤學 數學 cychen@mail.nctu.edu.tw

⼤學 數學 hlfu@math.nctu.edu.tw

⼤學數學 pyhuang@mail.ncku.edu.tw

⼤ ⼤學 數學 davidk@mail.ndhu.edu.tw

⼤學 tsfu@npic.edu.tw

⼤學 數學 deed@math.nccu.edu.tw

⼤學 數學 zhabiz@gmail.com

⼤學 數學 au4088@mail.au.edu.tw

⼤學 數學 huilan0102@gmail.com

⼤學數學 hsinhaolai@nknucc.nknu.edu.tw

⼤學 jsjuan@ncnu.edu.tw

⼤學數學 cmfu@mail.tku.edu.tw

⼤學數學 zhishi.pan@gmail.com

chaomingsun@gmail.com

⼤學 jlshang@mail.knu.edu.tw

⼤學數學 jackjunjiepan@gmai.com

⼤學 yllai@mail.ncyu.edu.tw

⼤學 數 chyen@mail.ncyu.edu.tw

⼤學 數學 kchuang@gm.pu.edu.tw

⼤學 jjlin@teamail.ltu.edu.tw

⼤學 birdy@teamail.ltu.edu.tw

⼤學 mjjou@teamail.ltu.edu.tw

研 學 研 數學 andanchen@gmail.com

博⼠ 研 ⼤學數學 huiqiulin@126.com

博⼠ 研 ⼤學 數學 yenpl.tw@gmail.com

博⼠ 研 ⼤學數學 hchsu0222@gmail.com

博⼠ 研 研 gaussla@gmail.com

博⼠ 研 研 數學 fyliao920@gmail.com

博⼠ 研 ⼤學數學 owlerson@yahoo.com.tw

博⼠ 研 ⼤學 數學 yphuang@isu.edu.tw

博⼠ ⼤學 數

博⼠ ⼤學 數學 tremolo.am96g@nctu.edu.tw

博⼠ ⼤學 數學 enix45@gmail.com

博⼠ ⼤學 數學 well.am94g@nctu.edu.tw

博⼠ ⼤學 數學 lienmy.am94g@nctu.edu.tw

博⼠ ⼤學 shuwen4477@yahoo.com.tw

博⼠ ⼤學數學 r95221034@ntu.edu.tw

學博⼠ ⼤學數學 zzuedujinyalei@163.com

學博⼠ ⼤學 數學 jinyu.97@gmail.com

學博⼠ ⼤學數學 amco0624@yahoo.com.tw

學博⼠ ⼤學 數學 ji3chinima@gmail.com

學博⼠ ⼤學 lu@math.cm.is.nagoya-u.ac.jp

學博⼠ ⼤學數學 jackhu@ntnu.edu.tw

學博⼠ ⼤學 數學 yjc7755@gmail.com

學博⼠ ⼤學 810111001@ems.ndhu.edu.tw

學博⼠ ⼤學 數學 jazztct@yahoo.com.tw

學博⼠ ⼤學 chlin@iastate.edu

學博⼠ ⼤學 程學 s96321901@gmail.com

碩⼠ ⼤學 數學 feel_1103@yahoo.com.tw

碩⼠ ⼤學 hmengping@gmail.com

碩⼠ ⼤學數學 jason32338@hotmail.com

碩⼠ ⼤學 數學 yuandy168@livemail.tw

碩⼠ ⼤學 數學 g039887905@gmail.com

碩⼠ ⼤學 數學 crazymars.am01g@nctu.edu.tw

碩⼠ ⼤學 數學 tim_lin30@yahoo.com.tw

博 碩⼠ ⼤學 數學 cktaiwanbear@hotmail.com

碩⼠ ⼤學 數學 vincentﬂame.am01g@nctu.edu.tw

碩⼠ ⼤學 數學 lilyamy04@hotmail.com

碩⼠ ⼤學 數學 doraemonzero0@gmail.com

碩⼠ ⼤學 數學 gfedc0606@gmail.com

碩⼠ ⼤學 數學 d23455432b@gmail.com

碩⼠ ⼤學數學 waterﬁrenopeace@hotmail.com

碩⼠ ⼤學 s1010449@mail.ncyu.edu.tw

碩⼠ ⼤學 程學 gm594@hotmail.com

碩⼠ ⼤學數學 Tiﬀ142842@gmail.com

碩⼠ ⼤學 johnmail5@gmail.com

碩⼠ ⼤學 數學 snowolf340@gmail.com

學碩⼠ ⼤學 sunbubbles1425@hotmail.com

學碩⼠ ⼤學數學

學碩⼠ ⼤學數學 christine_star19@hotmail.com

學碩⼠ ⼤學數學 leonyo83818@gmail.com

學碩⼠ ⼤學 數學 yaohisnhao@hotmail.com

學碩⼠ ⼤學 數學 genghao0720@gmail.com

學碩⼠ ⼤學 數學 jelha.ﬂas@gmail.com

學碩⼠ ⼤學 數學 beckppt.am02g@nctu.edu.tw

學碩⼠ ⼤學 數學 honge21129@gmail.com

學碩⼠ ⼤學 數學 yenjung0104@gmail.com

學碩⼠ ⼤學 數學 mimimandy135.am98@g2.nctu.edu.tw

學碩⼠ ⼤學 數學 simntimn@gmail.com

學碩⼠ ⼤學 數學 yses4131@gmail.com

學碩⼠ ⼤學 數學 MrBigTree1108@gmail.com

學碩⼠ ⼤學 數學 ddss12313@gmail.com

學碩⼠ ⼤學 tkst633@hotmail.com

學碩⼠ ⼤學 l123456578@hotmail.com

學碩⼠ ⼤學 程 s1000428@mail.ncyu.edu.tw

學碩⼠ ⼤學 程 s1020449@mail.ncyu.edu.tw

學碩⼠ ⼤學 數學 s1010244@mail.ncyu.edu.tw

學碩⼠ ⼤學 數學 mensa1888@gmail.com

學碩⼠ ⼤學 程學 andy79920@gmail.com

學碩⼠ ⼤學 程學 s102321517@mail1.ncnu.edu.tw

學碩⼠ ⼤學數學 r02221003@ntu.edu.tw

新 ⼤ 學 jack.birdyen@msa.hinet.net

學⼠ ⼤學數學 mark82_11@yahoo.com.tw

學⼠ ⼤學 hiwang123@gmail.com

學⼠ ⼤學

程 學⼠ ⼤學 magrady.24@gmail.com

28

### 新苗

研討會

21 2014.8.2–3 ⼤學數學 組合數學新苗研討會

20 2013.8.10–11 ⼤學數學 組合數學新苗研討會

19 2012.8.10–12 ⼤學 數學 組合研討會 組合數學新苗研討會

18 2011.8.6–7 ⼤學數學 組合數學 新苗研討會

17 2010.8.7–8 研 數學研 組合數學新苗研討會

16 2009.8.10–11 ⼤學 數學 組合數學 新苗研討會

15 2008.8.11–12 ⼤學 數學 組合數學 新苗研討會

14 2007.8.10–11 ⼤學 數學 組合數學 (新苗) 研討會

13 2006.8.11–12 ⼤學新 組合數學研討會

12 2005.8.26–27 ⼤學 數學 組合數學研討會

11 2004.8.27–28 ⼤學 數學 組合數學研討會

10 2003.9.5–6 ⼤學數學 組合數學研討會

9 2002.6.28–29 ⼤學數學 組合數學新苗研討會

8 2001.6.16–17 ⼤學數學 組合數學新苗研討會

7 2000.6.16 ⼤學 數學 組合數學新苗研討會

6 1999.6.11–12 研 數學 組合數學新苗研討會

5 1998.6.26–27 ⼤學 數學 組合數學新苗研討會

4 1997.6.27–28 ⼤學 數學 組合數學研討會

3 1996.5.24–25 ⼤學數學 組合數學研討會

2 1995.6.15–16 ⼤學 數學 組合數學新苗研討會

1 1990.6.25–26 研 數學研 數學新苗研討會

29

**Memo**

30

**Memo**

31

**Memo**

32