L打結數問題之研究

行政院國家科學委員會專題研究計畫 成果報告

l -打結數問題之研究 研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 99-2221-E-011-106-

執 行 期 間 ： 99 年 08 月 01 日至 100 年 07 月 31 日 執 行 單 位 ： 國立臺灣科技大學資訊管理系

計 畫 主 持 人 ： 王有禮

計畫參與人員： 碩士班研究生-兼任助理人員：王子欣 碩士班研究生-兼任助理人員：林冠宇 博士班研究生-兼任助理人員：林建宏 博士班研究生-兼任助理人員：郭俊麟

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 100 年 10 月 23 日

On Common Arcs of Cycles in Directed Chordal Rings

Ming–Tsan Juan¹, Yue–Li Wang^1,† and Cheng–Huang Hung¹

1 Department of Information Management,

National Taiwan University of Science and Technology, Taipei, Taiwan

Abstract

A digraph D is a Hamiltonian digraph if D has a directed Hamiltonian cycle; in this case, D is denoted byD. In a digraph D with n vertices, the number of common arcs between cycles Cn and C` is denoted by KD(Cn, C`). Let C(D, `) denote the set of all `-cycles in D. The maximum number of common arcs between Cn and all `-cycles in D is defined to be κ1(D, `, Cn) = max

x∈C(D,`){KD(Cn, x)} if there exists a C` in C(D, `) and ∞ otherwise.

The maximum number of common arcs between all Cn’s and `-cycles in D is defined to be κ2(D, `) = max

Cn∈D{κ1(D, `, Cn)}. If D is an orientation of an undirected graph G and there is a directed Hamiltonian cycle in D, then D is called a Hamiltonian digraph of G, denoted D(G). Let H denote the set of all Hamiltonian digraphs of G. The `-knot number of a graph G, denoted κ(G, `), is defined as min

x∈H{κ2(x, `)}. In this paper, we are concerned with the

`-knot number of chordal rings R(2t, {t}).

Keywords: Knot numbers, Hamiltonian digraphs, Chordal rings, Cycle pancyclism.

1 Introduction

Given a digraph D with n vertices, an `-cycle in D, denoted C`, is a cycle of ` distinct vertices.

A (directed) Hamiltonian cycle is a (directed) cycle of length n. A digraph D is called a Hamiltonian digraph if D has a directed Hamiltonian cycle; in this case, D is denoted byD. For a Hamiltonian digraphD, an arc in C`, ` < n, but not in C_n is called an alone arc with respect to C_n; otherwise, it is a common arc between them. The number of common arcs between C_n and C_{`</sub> inD is denoted by K<sub>D</sub>(Cn, C<sub>`}). Let C(D) denote the set of all `-cycles with ` < n in D

∗This work was supported in part by the National Science Council of the Republic of China under contracts NSC 97-2218-E-128-001- and NSC 97-2221-E-011-158-MY3.

†All correspondence should be addressed to Professor Yue–Li Wang, Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC(Email: ylwang@cs.ntust.edu.tw).

and C(D, `) denote the set of all `-cycles in D. The maximum number of common arcs between a specific Cn and all `-cycles in D is defined as

κ₁(D, `, Cn) =





 max

x∈C(D,`)

{K_D(C_n, x)} if C(D, `) 6= ∅,

∞ otherwise.

The maximum number of common arcs between all C_n’s and all `-cycles inD for a specific ` is defined as κ₂(D, `) = max

Cn∈D{κ₁(D, `, Cn)}.

Proposition 1 If there is exactly one directed Hamiltonian cycle inD, then κ2(D, `) = κ1(D, `, C_n).

Let G be an undirected graph with vertex set V (G) and edge set E(G). Hereafter, we use n and m to denote the cardinalities of V (G) and E(G), respectively. Let D be an orientation of G with vertex set V (D) = V (G) and arc set A(D). Note that if (x, y) is an edge in E(G), then ex- actly one of arcs hx, yi and hy, xi exists in A(D). For consistency, we also use (x0, x1, . . . , xk, x0) to denote a cycle in an undirected graph while hx0, x1, . . . , xk, x0i denotes a directed cycle in a digraph. If D has a directed Hamiltonian cycle, then D is called a Hamiltonian digraph of G, denotedD(G). Let H(G) denote the set of all Hamiltonian digraphs of G. If the context is clear, then we simply write D and H rather than D(G) and H(G), respectively. When mentioning more than one Hamiltonian digraph of G, we use Di, i = 1, 2, . . . , |H| to denote them. For example, see Figure 1. Figure 1(a) shows a graph G and Figure 1(b) depicts all Hamiltonian digraphs in H(G).

a

d c

(a) A graph G

a

d c a

d c

a

d c

a

d c

(b) All Hamiltonian digraphs in H(G)

Figure 1: An example to illustrate H(G).

Based on the definition of κ₂(D, `), the `-knot number of an undirected graph G, denoted κ(G, `), is defined as min

x∈H{κ₂(x, `)}. We use Figures 2 and 3 to demonstrate the above terms.

Figure 2(b) is a Hamiltonian digraph, sayD1, of Figure 2(a). We can see thatD1 has exactly one directed Hamiltonian cycle Cn= ha, b, c, d, e, f, g, ai. Furthermore, there also exist exactly one 3-, 4-, and 5-cycles inD1which are C3 = ha, b, g, ai, C4 = ha, e, f, g, ai, and C5= ha, d, e, f, g, ai, respectively. The values of K_D1(Cn, C3), K_D1(Cn, C4), K_D1(Cn, C5), and K_D1(Cn, C6) are 2, 3, 4, and ∞, respectively.

a b

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e d f

g

(a) A graph G

a b

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e d f

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(b) D1(G)

Figure 2: An example to illustrate C(D1, `).

Figure 3 depicts all Hamiltonian digraphs in H of Figure 2(a).

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HD7 HD8 HD9 HD10 HD11 HD12

HD13 HD14 HD15 HD16 HD17 HD18

HD19 HD20 HD21 HD22 HD23 HD24

HD25 HD26 HD27 HD28 HD29 HD30

Figure 3: An illustration of all Hamiltonian digraphs in H of Figure 2(a).

In Figure 3, there is also only one Hamiltonian cycle in D5. However, there are two 3- cycles in D5, namely ha, b, g, ai and ha, d, e, ai, and their corresponding K_D5(Cn, C3) are 2 and 1, respectively. Thus, κ1(D5, 3, Cn) = max{2, 1} = 2. InD15, there are two Hamiltonian cycles c1 = ha, b, c, d, e, f, g, ai and c2= ha, e, f, g, b, c, d, ai, one 6-cycle c3 = hb, c, d, e, f, g, bi, and two 4-cycles c4= ha, e, f, g, ai and c5 = ha, b, c, d, ai. We can see that there are no C3 and C5 inD15. Therefore, both κ₂(D15, 3) and κ₂(D15, 5) are ∞. The value of κ₂(D15, 4) is max{κ₁(D15, 4, c₁), κ₁(D15, 4, c₂)} = 3. The value of κ₂(D15, 6) is max{κ₁(D15, 6, c₁), κ₁(D15, 6, c₂)} = 5. Table 1 shows all values of κ₂(Di, `). We can see that κ(G, 3), κ(G, 4), κ(G, 5), and κ(G, 6) are 1, 3, 4, and 5, respectively.

Table 1: All values of κ₂(Di, `).

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10

C3 2 2 2 2 2 2 2 2 ∞ ∞

C4 3 3 3 3 ∞ ∞ 3 3 3 3

C5 4 4 ∞ ∞ 4 4 4 4 4 4

C₆ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 5 5

D11 D12 D13 D14 D15 D16 D17 D18 D19 D20

C3 1 1 ∞ ∞ ∞ ∞ 2 2 ∞ ∞

C4 ∞ ∞ 3 3 3 3 3 3 3 3

C5 4 4 4 4 ∞ ∞ ∞ ∞ 4 4

C6 5 5 5 5 5 5 ∞ ∞ 5 5

D21 D22 D23 D24 D25 D26 D27 D28 D29 D30

C₃ 2 2 ∞ ∞ 2 2 1 1 2 2

C4 3 3 3 3 3 3 ∞ ∞ ∞ ∞

C5 4 4 4 4 4 4 4 4 4 4

C6 ∞ ∞ 5 5 ∞ ∞ 5 5 ∞ ∞

The `-knot number problem was first studied by Galeana-Sanchez and Rajsbaum [7] in which they call the `-knot number problem the cycle-pancyclism problem. They also gave a lower bound of the `-knot number for tournaments with 3 6 ` 6 (n + 4)/2. Later on, they proposed the lower bounds of `-knot numbers for tournaments with larger cycles and bipartite tournaments [8,9,11,12]. Their results on the `-knot number problem are summarized in Table 2.

Note that a tournament is vertex-pancyclic [4,13]. Since a lot of graphs are not vertex-pancyclic, we modify the definition of the cycle-pancyclism problem as the `-knot number problem by considering κ<sub>1</sub>(D, `, Cn) = ∞ when digraph D does not have a cycle of length `.

Table 2: Previous results on `-knot numbers.

Graph classes (lower bounds of) `-knot numbers Tournaments [7] κ(G, `) = ` − 3, for 3 6 ` 6 ⁿ⁺⁴₂

Tournaments [8, 10] κ(G, `) > ` −²⁽ⁿ⁻³⁾<sub>n−`+3</sub> − 2, for <sup>n+4</sup><sub>2</sub> < ` < n Tournaments [9] κ(G, `) = ` − 4, if n 6 2` − 5

Bipartite tournaments [11] κ(G, `) = ` − 3, for 4 6 ` 6 <sup>n+4</sup><sub>2</sub> Bipartite tournaments [12] κ(G, `) > ` − 4, for <sup>n+6</sup><sub>2</sub> 6 ` 6 n − 2

A ring is a graph G = (V, E) with V = {x0, x1, . . . , xn−1} and E = {(x_i, x(i+1) mod n)|i = 0, 1, . . . , n − 1}. The edges in E are called ring edges. A chordal ring, denoted R(n, {l1, . . . , lt}) with l1 6 l2 6 . . . 6 lt, is an augmented ring where n is the number of nodes in the ring and l_i ∈ {2, . . . , bn/2c} connects every pair of nodes in the ring that are at distance l_i which is called a chord [5]. For some variations of chordal rings, the reader is referred to [1–3, 6, 14, 15]. In this paper, we are concerned with the `-knot number of a special chordal ring R(2t, {t}) (see Figure 4). In the next section, we shall derive the `-knot number for κ(R(2t, {t}), `). Note that the subscript of any vertex in a ring is taken modulo 2t.

x0x₁ x₂ x₃

xt−3 xt+1

xt+2

xt+3

x2t−2

x2t−3

xt−2

xt−1

x_t x2t−1

Figure 4: A chordal ring R(2t, {t}).

2 The `-knot number of R(2t, {t})

Theorem 2 For a chordal ring R(2t, {t}),

κ(R(2t, {t}), `) >















2 if ` = 3, 4,

t if t + 1 6 ` 6 2t − 1 and ` is an odd integer, min{` − 3, t, (3` − 2t)/2} if ` > 4 is an even integer,

∞ otherwise.

References

[1] W. Arden and H. Lee, Analysis of chordal ring Network, IEEE Transactions on Computers 30 (1981) 291–295.

[2] J.C. Bermond, F. Comellas, and D.F. Hsu, Distributed loop computer networks: a Survey, Journal of Parallel and Distributed Computing 24 (1995) 2–10.

[3] J.C. Bermond and C. Thomassen, Symmetry properties of chordal rings of degree 3, Dis- crete Applied Mathematics 129 (2003) 211–232.

[4] J.C. Bermond and C. Thomassen, Cycle in digraphs - A survey. Journal of Graph Theory 5 (1981) 1–43.

[5] F.T. Boesch and R. Tindell, Circulants and their connectivity, Journal of Graph Theory 8 (1984) 487–499.

[6] M. Flammini, G. Gambosi, and S. Salomone, Interval labeling schemes for chordal rings, Pre-Proceedings of Colloquium on Structural Information and Communication Complexity, Ottawa, 1994, pp. 16–18.

[7] H. Galeana-Sanchez and S. Rajsbaum, Cycle-pancyclism in tournaments I, Graphs and Combinatorics 11 (1995) 233–243.

[8] H. Galeana-Sanchez and S. Rajsbaum, Cycle-pancyclism in tournaments II, Graphs and Combinatorics 12 (1996) 9–16.

[9] H. Galeana-Sanchez and S. Rajsbaum, Cycle-pancyclism in tournaments III, Graphs and Combinatorics 13 (1997) 57–63.

[10] H. Galeana-Sanchez and S. Rajsbaum, A conjecture on cycle pancyclism in tournaments, Discussiones Mathematicae-Graph Theory 18 (1998) 243–251.

[11] H. Galeana-Sanchez, Cycle-pancyclism in bipartite tournaments I, Discussiones Mathematicae-Graph Theory 24 (2004) 277–290.

[12] H. Galeana-Sanchez, Cycle-pancyclism in bipartite tournaments II, Discussiones Mathematicae-Graph Theory 24 (2004) 529–538.

[13] J.W. Moon, On subtournaments of a tournament, Canadian Mathematical Bulletin 9 (1966) 297–301.

[14] J.H. Park and K.Y. Chwa, Recursive circulant: A new topology for multicomputer net- works, in: Proc. of International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’94), Kanazawa, Japan, 1994, pp. 73–80.

[15] A.L. Rosenberg, Theoretical research on networks: models and methodology, in: Sirocco97, D. Krizanc, D. Peleg (Eds.),The Fourth International Colloquium on Structural Information and Communication Complexity, Proceedings in Informatics, Carleton Scienti;c, Ascona, Switzerland, 1 (1997) 283–293.

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日期:2011/10/18

國科會補助計畫

計畫名稱: l -打結數問題之研究 計畫主持人: 王有禮

計畫編號: 99-2221-E-011-106- 學門領域: 計算機圖學

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99 年度專題研究計畫研究成果彙整表

計畫主持人：王有禮 計畫編號：99-2221-E-011-106- 計畫名稱：l -打結數問題之研究

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打結數問題是循環泛圈問題的推廣問題。一個有向圖 D 具有漢米爾頓有向循環則稱 D 為漢 米爾頓有向圖，對一個具有漢米爾頓性質的無向圖 G，我們定義 G 的 k 打結數為對每個由 G 方向化後的漢米爾頓循環圖 D 中所有長度為 k 的循環和每個漢米爾頓循環之間最大的共 用弧數，如果 D 中沒有長度為 k 的循環，則共用弧數為∞，再針對所有的漢米爾頓循環圖 中取最小的共用弧數，即為 G 的 k 打結數。

泛圈問題自 1966 年第一次被提出後，有很多相關問題的研究。打結數問題在探討漢米爾 頓循環和其他長度的有向循環之間的關係，因此可以應用在這類有向循環的問題上。