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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 |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
118 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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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 |
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統計 Statistics |
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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 |
參考文獻 References |
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