對於p>=2q，我們定義圖形Kp/q，設0，1，2，...，p-1是Kp/q的頂點，如果
ij是Kp/q圖形內的一個邊，則q<= |i-j| <=p-q。設圖形G同態到Kp/q，而滿足此條件最小的有理數p/q，我們記為Xc(G)。同理圖形Kp/q同態到G，而滿足此條件最大的有理數p/q，我們記為Wc(G)。圖形G如果所有的誘導子圖H都滿足Xc(H)=Wc(H)我們便稱圖形G是圓流完美圖。
在這篇論文中，我們表示哪些有理數p/q可以讓Kp/q的補圖是圓流完美圖。我們同時也證明如果轉換圖G(n，s)它的生成集內元素不超過3個的時候，則G(n，s)是圓流完美圖。

For p>=2q，let Kp/q be the graph with vertices 0，1，2，…，p-1 in which
i~j if q<=|i-j|<=p-q. The circular chromatic number Xc(G) of a graph G is the
minimum of those p/q for which G admits a homomorphism to Kp/q. The circular clique number Wc(G) of G is the maximum of those p/q for which Kp/q admits a homomorphism to G.. A graph G is circular perfect if for every induced subgraph
H of G we have Xc(H)=Wc(H). In this paper,we characterize those rational
numbers p/q for which the complement of Kp/q are circular perfect. We also prove
that if G(n，S) is a circulant graph whose generating set S has cardinality at most
3，then G(n，S) is circular perfect.

Contents
1 Introduction 2
1.1 Circular chromatic number………………………………....2
1.2 Circular clique number……………………..……………....3
1.3 Circular perfect graphs……………………..……………....3
1.4 Previous results…………………………………………….4
1.5 Result of this thesis………………………………………...5
2 Complements of circular complete graphs 6
2.1 Circular chromatic number of the complements of circular
complete graphs……………………………………………6
2.2 Circular clique number of the complements of circular
complete graphs……………………………………………7
2.3 Circular complete graphs which are circular perfect………7
3 Circulant graphs 10
3.1 Circulant graphs…………………………………………..10
3.2 Circulant graphs with |S| = 3 is circular perfect………….10

J. Bang-Jensen and J. Huang,
Convex-round graphs are circular-perfect,
J. Graph Theory, 40 (2002), 182-194.

M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas,
Progress on perfect graphs,
Math.Program., 97(2003), 405-422.

Guogang Gao and Xuding Zhu,
Star-extremal and lexicographic product,
Discrete Mathematics 152 (1996) , 147-156.

K. W. Lih, D. F. Liu and X. Zhu,
Star-extremal Circulant Graphs,
SIAM J. Discrete Math 11 (1998) , 330-339.

E. Steffen and X. Zhu,
Star chromatic number of graphs,
Combinatorica, 16 (1996) , 439-448.

L.E. TROTTER, Jr,
A class of facet producing graphs for vertex packing polyhedra,
Discrete Mathematics 12 (1975), 373-388.

S. Tu,
Star-extremal Circulant of Graphs,
April 2004. Master Thesis, National Sun Yat-sen University.

A. Vince,
star chromatic number of graphs,
J. Graph Theorey, 12 (1988) , 551-559.

B Xu,
two conjectures equivlent to the perfect graph conjecture,
Discrete Mathematics, 258(2002) 347-351.

B Xu,
On minimally circular imperfect graphs with a major vertex,
manuscript, 2004.

B Xu,
A family of minimally circular imperfect series-parallel graphs,
manuscript, 2004.

X. Zhu,
Graphs whose circular chromatic number equals the chromatic number, Combinatorica 19 (1999) , 139-149.

X. Zhu,
Circular perfect graphs}, Journal of Graph Theory,
48(2005), 186-209.

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

X. Zhu,
Perfect graph for generalized colouring-circular perfect graphs,
Graphs, morphisms and statistical physics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 63, 177--193. Amer. Math. Soc., Providence, RI, 2004.

X. Zhu,
An analogue of theorem for the circular chromatic number. {II},
Graphs Combin., 19(2003), 419-432.

X. Zhu,
An analogue of theorem for the circular chromatic number,
Proc. Amer. Math. Soc., 129(2001),2845-2852.