n階的互質圖是一個點集合為{1,2,...,n} = [1,n]，兩點相連若且唯若兩數互質的圖。

對任意n點圖G而言，G的質標為一個一對一函數ϕ從點集合V(G)映射到[1,n]，使得G上的每一條邊uv符合ϕ(u)和ϕ(v)互質。Entringer提出猜想：任何樹圖都具有質標。截至目前為止，這個猜想仍未被證實。藉由互質圖的定義，若我們能夠證明任何n階樹圖都是n階互質圖的生成子圖，即能驗證這個猜想。這激發我們研究互質圖的結構，當我們越認識互質圖，就能更廣泛地了解圖的質標，不僅僅是針對樹圖。

在本篇論文中，我們先發現一些互質圖的良好結構。我們證明了對於任意的正整數k，互質圖當點數足夠多時皆為k-連通。在著色問題中，我們證明了互質圖是弱完美圖和第一類圖。另外，我們計算了互質圖最小度數的上界，這讓我們知道一個圖若具有過大的最小度數，則必沒有質標。最後，互質圖的2-因子的存在性也是我們關注的問題之一。我們證明了至多四個連通部件的2-因子和由某些相同長度的環形圖組成的2-因子必為互質圖的生成子圖。

A coprime graph of order n is a graph whose vertex set is
{1,2,...,n} = [1,n] and two vertices are adjacent if and only if they are relatively prime, coprime in short.

A prime labeling of a graph G of order n is a bijection ϕ from V(G) onto [1,n] such that for every edge uv, gcd(ϕ(u),ϕ(v)) = 1. That is, ϕ(u) and ϕ(v) are coprime. We say a graph G is prime if there exists a prime labeling of G. It was conjectured by Entringer that every tree has a prime labeling. So far, the conjecture remains unsettled. By the definition of coprime graphs, the conjecture can be proved if we are able to show that every tree of order n is a spanning subgraph of the coprime graph of order n. This motivates us to study the structure of coprime graphs. The more we learn from the structure of coprime graph, the better we know about the prime labeling of graphs, not only trees.

In this thesis, we first discover several good structures of coprime graph. It is proved that coprime graphs are k-connected for all k depending on n, weakly-perfect in vertex coloring and Class 1 in edge-coloring. Furthermore, we determine an upper bound for the minimum degree of coprime graphs, which shows that a graph with large minimum degree is not prime. Finally, we focus on the existence of 2-factors of coprime graphs. For four cycles in which at most one is odd order or cycles with certain fixed order, we prove that they are spanning subgraphs of coprime graphs.

摘要 i
Abstract ii
致謝 iii
Contents iv
List of Figures vi
1 Introduction 1
1.1 Background 1
1.2 Notation and Definition 2
2 Structure of Coprime Graphs 8
2.1 Basic Properties 8
2.1.1 Maximum and MinimumDegree 8
2.1.2 Radius and Diameter 10
2.1.3 Independence Number 11
2.1.4 Clique Number 11
2.1.5 Connectivity 12
2.2 Properties of Colorings 13
2.2.1 Chromatic Number 13
2.2.2 Induced Cycles 14
2.2.3 Chromatic Index 15
2.3 Estimations 18
2.3.1 Estimation of Minimum Degree 18
2.3.2 Estimation of Size 20
3 Spanning Subgraphs 24
3.1 2-Factors with Multiple Components 24
3.2 Ck-factors 28
4 Conclusion 35
Bibliography 36

[1] Tom M Apostol. Introduction to analytic number theory. Springer Science & Business Media, 2013.
[2] Hung-Lin Fu and Kuo-Ching Huang. On prime labelings. Discrete Mathematics, 127(1-3):181–186, 1994.
[3] Joseph A. Gallian. A dynamic survey of graph labeling. Electronic Journal of combinatorics, 1(DynamicSurveys):DS6, 2018. 24:199–220, 1896.
[4] Jacques Hadamard. Sur la distribution des z´eros de la function ζ(s) et ses cons´ equences arithm´ etiques. Bulletin de la Societ´e mathematique de France,
[5] Godfrey H. Hardy and Edward M. Wright. An introduction to the theory of numbers. Oxford university press, 1979.
[6] Penny Haxell, Oleg Pikhurko, and Anusch Taraz. Primality of trees. Journal of Combinatorics, 2(4):481–500, 2011.
[7] Charles Jean de La Vall´ ee Poussin. Recherches analytiques sur la th´eorie des nombres premiers, volume 1. Hayez, 1897.
[8] John B. Rosser and Lowell Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6(1):64–94, 1962. 36
[9] Ralph G. Stanton and Ronald C. Mullin. Construction of room squares. The Annals of Mathematical Statistics, 39(5):1540–1548, 1968.
[10] A. Tout, Abdallah N. Dabboucy, and K. Howalla. Prime labeling of graphs. National AcademyScience Letters-India, 11:365–368, 1982.
[11] Arnold Walfisz. Weylsche Exponentialsummen in der neueren Zahlentheorie. VEB Deutscher Verlag der Wissenschaften, 1963