假設 $G$ 為一圖且 $e=ab$ 是 $G$ 裡的一條邊。長度為 $k$ 的
$G$-necklace，$N_k(G)$，建構如下: 令 $Q_1,Q_2,...,Q_k$ 為 $k$
個與 $G$ 同構的圖，其中 $e_i=a_ib_i$ 是 $Q_i$ 中對應於 $e=ab$
的邊。先取 $Q_1,Q_2,...,Q_k$ 的聯集 $Q_1
cup Q_2 cup cdots cup Q_k$，去掉邊 $e_1,e_2,...,e_k$ ，並增加一個頂點 $u$ 及邊 $ua_1,b_1a_2,b_2a_3,
cdots ,b_{k-1}a_{k},b_ku$ 所得的圖就是 $N_k(G)$。這篇論文求出所有 $K_{2n}$-necklaces 和所有 $K_{m,m}$-necklaces 的邊環色數。(見電子論文第五頁)

Suppose $G$ is a graph and $e=ab$ is an edge of $G$. For a
positive integer $k$, the $G$-necklace of length $k$ (with respect
to edge $e$), denoted by $N_k(G)$, is the graph constructed as
follows: Take the vertex disjoint union of $k$ copies of $G$, say
$Q_1 cup Q_2 cup cdots cup Q_k$, where each $Q_i$ is a copy of
$G$, with $e_i=a_ib_i$ be the copy of $e=a b$ in $Q_i$. Add a
vertex $u$, delete the edges $e_i$ for $i=1, 2, cdots, k$ and
add edges: $ua_1, b_1a_2, b_2a_3, cdots, b_{k-1}a_k, b_ku$. This
thesis determines the circular chromatic indexes of $G$-necklaces
for $G = K_{2n}$ and $G= K_{m, m}$.(見電子論文第六頁)

1 Introduction 2
1.1 Definitions…………………………….... 2
1.2 Some known results……………………. 3
1.3 Main results of the thesis……………….. 6
2 The Km,m-Necklaces 8
2.1 Notation and lemmas……………………. 8
2.2 Proving the upper bound………………… 9
2.3 Proving the lower bound…………………11
2 The K2n-Necklaces 14
2.2 Proving the lower bound………………… 14
2.3 Proving the upper bound………………… 17

[1]P. Afshani, M. Ghandehari, M. Ghandehari, H. Hatami, R.
Tusserkani and X. Zhu, Circular Chromatic Index of Graphs of
Maximum Degree 3, J. Graph Theory, to appear.

[2]D.R. Guichard, Acyclic graph coloring and the complexity of
the star chromatic number, J. Graph Theory, 17(1993), 129--134.

[3] T. Kaiser, D. Kral and
R. Skrekovski,
A revival of the girth conjecture,
J. Combin. Theory Ser. B, 92(2004), 41--53.

[4] T. Kaiser, D. Kral, R. Skrekovski and X. Zhu,
The circular chromatic index of graphs of large girth,
manuscript, 2004.

[5] A. Nadolski, The circular chromatic index of some Class 2
graphs, Discrete Mathematics, to appear.


[6] A. Vince, Star chromatic number, J. Graph Theory,
23(1988), 551--559.

[7] D. B. West, Introduction to graph theory,
Prentice Hall Inc., Upper Saddle River, NJ, 1996.


[8] D.B. West and X. Zhu, Circular chromatic index of the Cartesian
product of graphs, manuscript, 2004.

[9] X. Zhu, Circular chromatic number, a survey,
Discrete Mathematics, 229 (2001), 371-410.