本文探討定向圖上的相依邊(dependent arc)的相關性質。給定圖 G
的無圈定向圖 D，Edelman [31] 給出以下定義: 若邊倒轉方向後會
產生一有向圈(directed cycle)，則稱此邊為相依邊(dependent arc)，否則稱此邊為獨立邊(independent arc)。令 d(D) 代表 D
上的相依邊總數，i(D) 代表 D 上的獨立邊總數，
d_{max}(G)(d_{min}(G)) 代表 G 的所有無圈定向圖中所能給出的最多(最少)相依邊數，i_{max}(G) (i_min}(G)) 代表 G
的所有無圈定向圖中所能得到的最多(最少)獨立邊數。Edelman [6]
證明了當 G 為連通圖時，可得 d_{max}(G) = ||G||-|G|+1。如果圖
G 存在有無圈定向圖使得其相依邊數可為 d_{min}(G) 到
d_{max}(G) 中的任何整數，則稱圖 G 具有內插性質(interpolation
property)或稱之為可完全定向的(fully orientable)。West
證明了完全二部圖(complete bipartite graph)具有內插性質[31]。
我們確認了外向平面圖(outerplane graph)的最少相依邊數，
並證明了外向平面圖具有內插性質。令 N(G) 代表收集圖 G
上的所有無圈定向圖所能獲得的獨立邊數所成集合，並稱之為 G
的獨立邊譜(independent arc spectra)。我們求得完全 k
部圖(complete k-partite graph) G 的最少獨立邊數，
並刻劃某些圖類的獨立邊譜。覆蓋圖(cover
graph)是一種有限偏序集(finite partially ordered
set)的漢斯圖(Hasse diagram)的底圖(underlying
graph)。覆蓋問題(cover problem)是判斷所給圖是否為覆蓋圖的問題。明顯可得 G 為一覆蓋圖若且唯若 d_{min}(G)=0。我們證明了當 m geq 1、t geq 1、k geq 3、及 n geq 2k+2 時，奇圈(odd cycle) C_{2t+1} 的廣義 Mycielski 圖 {sf M}_m(C_{2t+1})、Kneser 圖 {sf KG}(n, k)、與 Schrijver 圖 {sf SG}(n,k) 都不是覆蓋圖。

In this thesis, we focus on the study of dependent arcs of
acyclic orientations of graphs. Given an acyclic orientation D of G, Edelman cite{West} defined an arc to be {em dependent} if its reversal creates a cycle in D; otherwise, it is independent. Let d(D) and i(D) be the numbers of dependent arcs and independent arcs in D, respectively. And, let d_{max}(G)(d_{min}(G)) and i_{max}(G) (i_{min}(G)) be the maximum (minimum) numbers of dependent arcs and independent
arcs over all acyclic orientations of G, respectively. Edelman
cite{Fisher} showed that if G is connected, then
d_{max}(G)=||G||-|G|+1. A graph G is said to satisfy the
{em interpolation property} (or G is fully orientable) if $G$
has an acyclic orientation with exactly k dependent arcs for
every k with d_{min}(G) leq k leq d_{max}(G). West
established the interpolation property for complete bipartite graphs cite{West}. We obtain the minimum numbers of dependent arcs of the outerplane graphs and show that the outerplane graphs satisfy the interpolation property. Let N(G) be the set { i(D)| D is an acyclic orientation of G }. N(G) is called the independent-arc spectra of G. For complete k-partite graphs G, we obtain i_{max}(G) and discuss the independent-arc spectra for some classes. On the other hand, we consider the cover problem. A cover graph is the underlying graph of the Hasse diagram of a finite partially ordered set. The cover problem is that whether a given graph is a cover graph. It is easy to see that a graph G is a cover graph if and only if d_{min}(G)=0. We show that the generalized Mycielski graphs M_m(C_{2t+1}) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m geq 1, t geq 1, k geq 3 and n geq 2k+2.

Contents
摘要
Abstract
誌謝
Contents
List of Figures

Chapter 1 Introduction 1
1.1 Basic Notation……………………………………………..............1
1.2 Interpolation Property……………………………………………..3
1.3 Cover graphs……………………………………………................6

Chapter 2 Outerplane graphs and Interpolation Property 11
2.1 Maximal outerplane graphs………………………………………11
2.2 Outerplane graphs………………………………………………...15

Chapter 3 The Independent-arc Spectra of Complete k-partite Graphs 18
3.1 The Numbers of Maximum Independent Arcs of Complete k-partite Graphs………………………………………………....................18
3.2 The Independent-arc Spectra of Balanced Complete k-partite Graphs…………………………………………………………....21
3.3 The Interpolation Property of Some Unbalanced Complete k-partite Graphs………………………………………………....................27

Chapter 4 Non-cover Generalized Mycielski, Kneser, and Schrijver Graphs
31
4.1 The Generalized Mycielski Graphs of Cycles………………….. 31
4.2 Tower Graphs……………………………………………………33
4.3 Kneser Graphs…………………………………………………...36
4.4 Schrijver Graphs…………………………………………………40
4.5 Application to Circular Chromatic Number……………..............41

Chapter 5 Further Research 42

Bibliography………………………………………………………………43

C. Berge, Two theorems in graph theory,
Proc. Nat. Acad. Sci. U. S. A. 43, 842 - 844, 1957.

G. Birghtwell, On the complexity of diagram testing, Order 10, 297-303, 1993.

G. Chartand and L. Lesniak, Graphs and Digraphs, 3rd
edition, 1996.

Karen L. Collins and Kimberly Trsdal, Dependent
edges in mycielski graphs and 4-colorings of 4-skeletons, J.
Graph Theory 46, no. 4, 285-296, 2004.

P. Edelman, private communication.

D. C. Fisher, K. Fraughnaugh, L. Langley, and D. B. West, The number of dependent arcs in an acyclic orientation, J. Combin. Theory Ser. B 71, 73-78, 1997.

T. Gallai, Uber extreme Punkt- und Kantenmengen,
Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 2, 133 - 138, 1959.

L. A. Goddyn, M. Tarsi, and C. Q. Zhang,
On (k,d)-colorings and fractional nowhere-zero flows, J. Graph Theory, 28, 155--161, 1998.

A. Johnson, F. C. Holroyd, and S. Stahl,
Multichromatic numbers, Star chromatic numbers and Kneser graphs, J. Graph Theory, 26(3): 137-145, 1997.

W. Lin and P. C. B. Lam, Circular Chromatic numbers of the
generalized Mycielskians of cycles, preprint.

W.-T. Li and K.-W. Lih, The circular extremality of
some reduced Kneser graphs, preprint, 2003.

Wei Ping Liu and Ivan Rival, Inversions, cuts, and
orientations, Discrete Mathematics 87, 163-174, 1991.

L. Lovasz, Kneser's conjecture, chromatic number
and homotopy, J. Combin. Theory, Ser. A, 25, 319--324, 1978.

K. M. Mosesian, Some theorems on strongly basable
graphs, Akad. Nauk. Armian SSR. Dokl., 54, 241--245 (in Russian), 1972.

J. Mycielski, Sur le coloriage des graphes, Coll. Math.,
3, 161--162, 1955.

J. Nevsetvril and V. Rodl, More on complexity of the
diagrams, comment. Math. Univ. carolinae 2, 269-278, 1995.

O. Pretzel, A Non-covering Graph of Girth Six,
Discrete Mathematics, 63, 241-244, 1987.

O. Pretzel, On graphs that can be oriented as diagrams of
ordered sets, Order, 2, 25-40, 1985.

O. Pretzel, On reorienting graphs by pushing down
maximal vertices, Order, 3, 135-153, 1986.

O. Pretzel, Orientations and edge functions of graphs,
in A. D. Keedwell(ed.), Surveys in Combinatorics, CUP, LMS Lecture Notes 166, pp. 161-185, 1991.

O. Pretzel, Orientations and reorientations of graphs,
Contemporary Mathematics Volumn 57, 103-125, 1986.

O. Pretzel, Orientations of Chain Groups,
Order, 12, 135-147, 1995.

O. Pretzel and D. Young, Balanced graphs and
noncovering graphs, Discrete Math. 88, 279-287, 1991.

O. Ore, Theory of graphs, Amer. Math. Soc.
Coll. 34, Providence, R. I. 1962.

I. Rival, The diagram, Graphs and Order
(Banff, Alta., 1984), 103--133, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 147, Reidel, Dordrecht, 1985.

Carla D. Savage and Cun-Quan Zhang, The
connectivity of acyclic orientation graphs, Discrete Mathematics 184, 281-287, 1998.

A. Schrijver, Vertex-criticl subgraphs of
Kneser graphs, Nieuw Arch. Wisk. (3), 26(3):451-461, 1978.

G. Simonyi and G. Tardos, Local chromatic number, Ky Fan's
theorem, and circular colorings, arXiv:math.CO/0407075 v3 26 Nov 2004.

C. Tardiff, Fractional chromatic numbers of cones
over graphs, J. Graph Theory, 38, 87--94, 2001.

J. Topp and P. D. Vestergaard,
Interpolation theorems for domination numbers of a graph,
Discrete Math. 191, 207 - 221, 1998.

D. B. West, Acyclic orientations of complete bipartite graphs, Discrete Math. 138, 393 - 396, 1995.

D. B. West, Introduction to Graph Theory, First
Edition, Prentice Hall, Upper Saddle River, 1996.

X. Zhu, Circular chromatic number: a survey,
Discrete Math. 229, 371-410, 2001.