臺灣博碩士論文加值系統

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

[1] Guogang Gao and Xuding Zhu,Star-extremal graphs and lexicographic product, Discrete Mathematics 152(1996),147-156
[2] H.L. Abbott and Xuding Zhu, The star chromatic number of a graph, J. Graph Theory 17(1993),394-360
[3] A. Vince , Star chromatic number, J. Graph Theory,12,(1988),pp.551-559
[4] k.W.Lih,D.F.Liu and X.Zhu,Star-extremal Circulant Graphs,SIAM J. Discrete Math 11(1998),330-339.