就圖而言，如果它的分數著色數與其環圍著色數相同，則我們稱此圖為star extremal。給定正整數n,k,k' 使得 1<=k<=k'<=n/2 則循環圖的頂點集為[n]={0,1,2,...,n-1} ，在這當中頂點i 和頂點j 相連的條件為：k<=|i-j|<=k'，或者n-k'<= |i-j|<=n-k 。已知對於n=q(k+k')+r 其中0<=r <k+k'，如果k'>=5/4k 那麼 G(n,S_k,k')就是star extremal。在這篇論文裡，我們將證明：對於n=q(k+k')+r 其中0<=r <k+k' ，如果 k'>=7/6k而且q>=4，那麼G(n,S_k,k') 是star extremal。

A graph is called star extremal if its fractional chromatic number is equal to its circular chromatic number. Given integers n,k,k' such that 1<=k<=k'<=n/2,the circulant graph G(n,S_k,k') has vertex set [n]={0,1,2,...,n-1} in which i~j if k<=|i-j|<=k' or n-k'<= |i-j|<=n-k. It was known that for n=q(k+k')+r,where 0<=r <k+k', if k'>=5/4k,then G(n,S_k,k') is star extremal. In the thesis, we prove that if k'>=7/6k and q>=4, then G(n,S_k,k') is star extremal.

1.Introduction
1.1 Definition of X_c and X_f
1.2 Circulant graphs
1.3 Star extremal
1.4 Some known results
1.5 Main result of this thesis
2.Proof of the main result
2.1 Preliminary lemmas
2.2 Main proof
2.3 Examples

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