An Upper Bound for the Circular Chromatic Number of Mycielski Graphs 郭玟伶、黃鈴玲
E-mail: [email protected]
ABSTRACT
In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski ([15]) developed a graph transformation that transforms a graph G into a new graph M(G), we now call the Mycielskian of G. For t>= 2, Mt(G) = M(Mt-1(G)). The problem of determining the circular chromatic numbers of these graphs has been investigated in many papers. In this thesis, we shall study the range of Xc(Mt(G)), especially when G is a complete graph (Kn) or a circular complete graph Kkd. In [3], Chang, Huang, and Zhu proved that if ?偛c(G)<=X(G)-r with r=1/2 or 1/3, then Xc(M2t(G))<=(M2t(G))-r for every positive integer t. We find that this property is also true for r = 2/3. That is, when ?偛c(G) is close to X(G)-1,Xc(M2t(G)) will also be close to ?偛(M2t(G))-1 for every positive integer t.
Keywords : Mycielski graph, circular chromatic number
Table of Contents
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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