本文選題：幻陣 + Heffter陣列�。� 參考：《北京交通大學》2017年碩士論文

【摘要】：用組合設計方法研究完全圖在曲面上嵌入是組合設計和圖論的重要研究課題.在定向曲面上完全圖K2mn+1的二嵌入是一個面2-染色的拓撲嵌入,并使得第一個顏色類構成一個s-圈,另一個顏色類構成一個t-圈[2].Grannel等人[11-13]研究了當s = t= 3時即Steiner三元系(STS)的二嵌入.2010年,Brown[7]構造了一類二嵌入,這類二嵌入使得s,t 一個等于3,另一個等于4.Archdeacon[2]在2014年提出的Heffter陣列的概念推廣了 Steiner三元系的二嵌入.Heffter陣列由兩個正交的Heffter系構成,從組合設計角度看,一個Heffter系可以構造一個循環(huán)的k-圈系.Heffter陣列與完全二部圖Km,n流量分配相關,滿足一定條件的流量圖可以用來構造完全圖可定向嵌入.所以,Heffter陣列可以用來構造在可定向曲面上完全圖的二嵌入.同年,Archdeacon與Dinitz等人[5]定義了帶空位置的整型Heffter方陣H(n;k).他證明H(n;k)存在的必要條件是nk≡ 0,3(mod 4)并猜想這個條件是充分的,并且構造了當k為偶數(shù)和nk≡3(mod4)時的H(n;k),對nk≡1(mod 4)給出了部分參數(shù)的構造.2015年,Dinitz和Mattern[2]構造了兩類不帶空的Heffter陣列,即H(3,n)和H(5,n).進一步地,Boothby[6]在他的博士畢業(yè)論文中證明了不帶空的Heffter陣列存在當且僅當m2,n2.本文主要介紹Heffter陣列和圖的二嵌入關系,并構造了一些帶空位置的整型Heffter方陣,一共分為四章.第一章,首先介紹有關的基本概念和符號.第二章,介紹了 Heffter陣列和圖的二嵌入的關系.第三章,在Archdeancon構造的帶空位置的整型Heffter方陣的基礎上,構造了某些參數(shù)的帶空位置的整型Heffter方陣.第四章,對所做的工作進行總結.
[Abstract]:Using the combined design method to study the complete graph embedding on the surface is an important research topic in combination design and graph theory. The two embedding of complete graph K2mn+1 on the directional surface is a topological embedding of a surface 2- dyeing, and the first color class constitutes a s- circle, and the other color class constitutes a t- circle [2].Grannel et al. [11-13] to study when s = t= 3, that is, the two embedded.2010 years of the Steiner three unit (STS), Brown[7] constructs a class of two embeddedness, and this kind of two embedding makes s, t one equal to 3, the other equal to the concept of Heffter array that 4.Archdeacon[2] put forward in 2014. The two embedded.Heffter array of Steiner three is composed of two orthogonal Heffter systems, from combination design. In view, a Heffter system can construct a cyclic k- loop.Heffter array which is related to complete two partite graph Km, n traffic distribution. The flow graph satisfying certain conditions can be used to construct complete graph orientable embeddable. So, Heffter arrays can be used to construct a complete graph on the orientable surface for two embedding. In the same year, Archdeacon and Dinitz etc. Human [5] defines an integer Heffter matrix H (n; K) with an empty position. He proves that the necessary condition for the existence of H (n; K) is NK 0,3 (MOD 4) and conjectured that the condition is sufficient and constructs the construction of the two types of non empty spaces when the K is the even number and the 1 (4). Heffter arrays, that is, H (3, n) and H (5, n). Further, Boothby[6] demonstrated in his doctoral thesis that the non empty Heffter array exists when and only as m2, n2. this article mainly introduces the two embeddedness of Heffter array and graph, and constructs a number of integer Heffter squares with empty positions, which are divided into four chapters. The basic concepts and symbols. In the second chapter, the two embedded relationship between Heffter array and graph is introduced. The third chapter, on the basis of the integral Heffter matrix with the empty position in Archdeancon, constructs the integral Heffter square of some parameters with the empty position.

【學位授予單位】：北京交通大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O157.2
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本文編號：1800040