Mycielski 圖形的環著色數上界 郭玟伶、黃鈴玲
E-mail: 9509866@mail.dyu.edu.tw
摘 要
在尋找不含三角形且有任意大的著色數圖形時，Mycielski([15]) 發展出一種圖形轉換方式，可以將一個圖形G變形成一個新 圖形 M(G)，稱之為G的Mycielskian；重複此轉換，當t>=2時，可以令Mt(G) = M(Mt-1((G))。關於這些圖形之環著色數 值(circular chromatic number; χc)的研究已經有一些發展。在這篇論文裡，我們主要探討 的是χc(Mt(G))可能的範圍，特別 是當G是完全圖(complete graph; Kn) 或環完全圖(circular complete graph;Kkd)時。Chang、Huang及Zhu 等人在[3]中證明了當 χc(G)<=χ(G)-r且r = 1/2或1/3時，可得到對任 意正整數t，χc(M2t(G))也會有<=χ(M2t(G)-r)的性質。我們進一步證 明了 此一性質在r = 2/3時也會成立；換言之，當χc(G)接近χc(G)-1 時，χc(M2t(G))也會接近χc(M2t(G))-1。
關鍵詞 : Mycielski 圖，環著色數
目錄
封面內頁 簽名頁 授權書...iii 中文摘要...iv 英文摘
要...v 誌謝...vi 目錄...vii 圖目 錄...ix 表目錄...x 1. Introduction 1.1. Basic definitions in graph theory ...1 1.2. Circular chromatic number...1 1.3. Mycielski graph ...3 2.
Preliminary results of Xc(M (G)) 2.1 Graphs G with Xc(M (G)) < X(M (G)) ...6 2.2 Graphs G with Xc(M (G)) = X(M (G)) ...7 2.3 Generalized Mycielski＊s graphs ...8 3. Upper bounds of d when Xc(Mt(G)) = k/d 3.1 The range of d when Xc(Mt(Kn)) = k/d ...11 3.2 The range of d when Xc(Mt( kd K )) = k/d ...12 4. The graphs Mt( kd K ) 4.1.
Graphs G with Xc(G) <= (X(G)-1)+1/3 ...16 4.2. Graphs G with Xc(G) >= X(G)-1/3 ...19 5. Conclusions ...20 Reference ...21
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