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博碩士論文 etd-0014114-151402 詳細資訊
Title page for etd-0014114-151402
論文名稱
Title
線圖和全圖的圓環染色
The circular chromatic numbers of line graphs and total graphs
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
118
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2013-11-24
繳交日期
Date of Submission
2014-01-14
關鍵字
Keywords
線圖、圓環染色、圓全色數、圓邊色數、全圖
line graph, circular colouring, circular chromatic index, circular total chromatic number, total graph
統計
Statistics
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The thesis/dissertation has been browsed 5726 times, has been downloaded 983 times.
中文摘要
圖G的一個圓環r染色(circular r-colouring)是一個函數c:V(G)→[0,r)使得對於任意兩個有邊相連的頂點x,y,滿足1≤|c(x)-c(y) |≤r-1。一個圖的圓色數χ_c (G)是指最小的r使得G存在一個圓環r染色。

圖G的一個圓環r邊染色(circular r-edge-colouring)是在G的線圖(L(G) )上的一個圓環r染色。一個圖的圓邊色數〖χ^'〗_c (G)=χ_c (L(G) )。已知對r∈(2,3],存在一個圖G的圓邊色數為r若且唯若存在整數k,r=2+ 1/k。在[23],Lukot^' ka和Maza ́k證明對於任意有理數r∈(3,10/3),存在一個有限圖G,它的圓邊色數為r。我們證明了如果k≥3為一奇數,則對於任意的有理數r∈(k,k+1/4),存在一個有限圖G,其圓邊色數為r(推論3.1.3);當k≥4為一偶數時,對於任意的有理數r∈(k,k+1/6),存在一個有限圖G,其圓邊色數為r(推論3.1.4)。

圖G的一個圓環r全染色(circular r-total-colouring)是在G的全圖(T(G) )上的一個圓環r染色。一個圖的圓全色數〖χ'〗_c (G)=χ_c (T(G) )。在[10],已知對r∈(3,4],存在一個圖G的圓全色數為r若且唯若存在整數k,r=3+ 1/k。我們證明了當n≥5為一整數,對於任意的有理數r∈(n,n+1/3),存在一個有限圖G,其圓全色數為r(定理3.2.2)。
Abstract
A circular r-colouring of a graph G is a mapping c : V (G) → [0, r) such that for any two adjacent vertices x and y, 1 ≤ |c(x) − c(y)| ≤ r−1. The circular chromatic number of G is χc(G) = inf{r : G has a circular r-colouring}.

A circular r-edge-colouring of a graph G is a circular r-colouring of its line graph L(G). The circular chromatic index, written χ′c(G), is defined by χ′c(G) = χc(L(G)). It is known that for r ∈ (2, 3], there is a graph G with χ′c(G) = r if and only if r = 2 + 1/k for k ∈ N. In [23], Lukot’ka and Maz´ak proved that for any rational r ∈ (3, 10/3), there is a finite graph G with χ′c(G) = r. We prove that if k ≥ 3 is an odd integer, then for any rational r ∈ (k, k + 1/4), there is a finite graph G with χ′c(G) = r (Corollary 3.1.3); if k ≥ 4 is an even integer, then for any rational r ∈ (k, k + 1/6), there is a finite graph G with χ′
c(G) = r (Corollary 3.1.4).

A circular r-total-colouring of a graph G is a circular r-colouring of its total graph T(G). The circular total chromatic number, written χ′′c(G), is defined by χ′′c (G) = χc(T(G)). In [10], it was proved that for r ∈ (3, 4], there is a graph G with χ′′c (G) = r if and only if r = 3 + 1/k for k ∈ N. We prove that for any integer n ≥ 5 and any rational r ∈ (n, n + 1/3), there is a finite graph G with χ′′c (G) = r (Theorem 3.2.2).
目次 Table of Contents
致謝i
摘要ii
Abstract iii
Contents iv
List of Figures vi
1 Introduction 1
1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Circular chromatic number . . . . . . . . . . . . . . . . . . . . 5
1.3 Circular chromatic numbers of line graphs . . . . . . . . . . . 10
1.4 Circular chromatic numbers of total graphs . . . . . . . . . . . 17
2 Building blocks 21
2.1 p-line n-regular monochromatic network for line graph . . . . 21
2.2 (p, n)-monochromatic network for total graph . . . . . . . . . 32
3 Construction method and main results 37
3.1 Construction of graphs with given circular chromatic indices . 37
3.2 Construction of graphs with given circular total chromatic
numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Tension50
4.1 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Lemmas of line graphs with n is odd . . . . . . . . . . . . . . 55
4.3 Lemmas of line graphs with n is even . . . . . . . . . . . . . . 65
4.4 Lemmas of total graphs . . . . . . . . . . . . . . . . . . . . . . 67
5 ϵ-changeability of blocks 71
5.1 ϵ-changeability of networks for line graph . . . . . . . . . . . . 71
5.2 ϵ-changeability of networks for total graph . . . . . . . . . . . 80
6 Improve the result on circular chromatic indices for even
regular graphs 84
6.1 Circular chromatic indices of even regular graphs . . . . . . . 84
7 Conclusions and further research 100
7.1 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 100
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