【作者】 张博；

【导师】 杨卫华；

【作者基本信息】 太原理工大学 ， 数学， 2018， 硕士

【摘要】 在这篇文章中,我们定义了有向图的生成连通度,推广了竞赛图的弱哈密尔顿连通性,研究了竞赛图的生成连通性,同时在二部竞赛图上研究了特殊圈的存在性,均得到了一些有意义的结果.全文结构如下:第一章,我们主要介绍了本文的研究背景,系统地阐述了生成连通性和圈结构的研究状况,进而提出研究的问题并给出相关结果.第二章,研究了竞赛图的生成连通性.我们定义了有向图的生成连通度并得到以下主要结果:(1)当k ≥ 0时,一个(2k + 1)-强连通的竞赛图是(k + 2)*-弱连通的.(2)当≥ 2时,一个2k-强连通的竞赛图是k*-强连通的.(3)在含有n个顶点的竞赛图中,它的不规则度为i(T)≤ k.如果n ≥ 6t + 5k(t ≥2),则κs*(T)≥t;如果n ≥ 6t + 5k-3(t ≥ 2),则κw*(T)≥ t + 1.第三章,研究了二部竞赛图的特殊圈的存在性.我们用归纳法定义了可分解的k-正则二部竞赛图并得到以下主要结果:可分解的k-正则二部竞赛图BT4k(k≥3)包含D(4k,p)对所有的2≤p≤4k成立,除非BT4k同构于一个有向图D:它有一个哈密尔顿圈(1,2,3,...,4k,1),对于任何点i∈(1,2,3,...,4k,1),满足条件(4m+i-1,i)∈ A(D)和(i,4m + i + 1)∈ A(D),其中1 ≤ m ≤ k-1,每个点i取模4k同余使得点4k+i就是点i.更多还原

【Abstract】 In this paper,we define the spanning connectivity of digraphs,generalize the weak Hamil-tonian connectedness of tournaments,and study the spanning connectivity of tournaments.We also study the existence of specified cycles in bipartite tournaments.We have got some meaningful results.The overall structure of this paper is as follows:In Chapter 1,we mainly introduce the research background of this paper,and system-atically expound the research status of spanning connectivity and cycle structure.Then we put forward the problems and give the related results in this paper.In Chapter 2,we study the spanning connectivity of tournaments.We define the spanning connectivity of digraphs and get the following main results:(1)for k≥ 0,a(2k + 1)-strong tournament is(k + 2)*-weakly connected.(2)for k≥ 2,a 2k-strong tournament is k*-strongly connected.(3)In a tournament with n vertices and irregularity i(T)≤k,if n ≥ 6t + 5k(t ≥ 2),then ks*(T)≥ t;if n ≥ 6t + 5k-3(t ≥ 2),then kw*(T)≥ t + 1.In Chapter 3,we study the existence of specified cycles in bipartite tournaments.We define the decomposable k-regular bipartite tournament by induction and get the following main results:a decomposable k-regular(k ≥ 3)bipartite tournament BT4k contains D(4k,p)for 2<p<4k unless BT4k is isomorphic to a digraph D which has a Hamiltonian cycle(1,2,3,...,4k,1),for any vertex i∈(1,2,3,...,4k,1),there are(4m + i-1,i)E A(D)and(i,4m + i + 1)E A(D),where 1 ≤ m ≤ k-1,every vertex i modulo 4k so that the vertex 4k + i is the vertex i.更多还原