3-Factor-criticality of vertex-transitive graphs

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if and only if it is not a cycle.Comment: 15 pages, 3 figure

On the connectivity of graphs

本文以点传递图和双轨道图为中心，分别以图的限制和超限制边连通性、圈边连通性为主题展开对着两类图的研究，尤其对双轨道图的研究较为系统全面，该部分结果主要集中在第二章至第四章，具体来讲，在第二章里我们研究了点传递图以及极小Cayley图的超2-限制和超3-限制边连通性，较完整的刻画了超2-限制边连通的点传递图和极小Cayley图，同时我们给出了点传递图是超3-限制边连通的一个充分条件；在第三章里我们研究了等阶双轨道图的边连通性，限制边连通性，超边连通性；第四章里我们研究了等阶双轨道图和正则双轨道图的全边连通性；第五章里我们研究了部分最小限制k-边连通图；第六章研究了两类网络的度限制连通性；最后，我...In a network, traditional connectivity is an important measure since it can correctly reflect the fault tolerance of network systems with few processors. However, it always underestimates the resilience of large networks. There is a discrepancy because the occurrence of events which would disrupt a large network after a few processor or link failures is highly unlikely. Thus the disruption en...学位：理学博士院系专业：数学科学学院数学与应用数学系_应用数学学号：1902009015361

3-restricted edge connectivity of vertex transitive graphs

A 3-restricted edge cut is an edge cut that disconnects a graph into at least two components each having order at least 3. The cardinality lambda(3) of minimum 3-restricted edge cuts is 3-restricted edge connectivity. Let G be a connected k-regular graph of girth g(G) at least 4 and order at least 6. Then lambda(3) <= 3k - 4. It is proved in this paper that if G is a vertex transitive graph then either lambda(3) = 3k - 4 or lambda(3) is a divisor of vertical bar G vertical bar such that 2k - 2 <= lambda(3) <= 3k - 5 unless k = 3 and g(G) = 4. If k = 3 and g(G) = 4, then lambda(3) = 4. The extreme cases where lambda(3) = 2k - 2 and lambda(3) = 3k - 5 are also discussed

3-Restricted Edge Connectivity of Vertex Transitive Graphs of Girth Three

设G是阶至少为6的k正则连通图.如果G的围长等于3,那么它的3限制边连通度 λ3(G)≤3k-6.当G是3或者4正则连通点可迁图时等号成立,除非G是4正则图并且 λ3(G)=4.进一步,λ3(G)=4的充分必要条件是图G含有子图K4.Let G be a k-regular connected graph of order at least six. If G has girth three, its 3-restricted edge connectivity λ3(G) ≤3k-6. The equality holds when G is a cubic or 4-regular connected vertex-transitive graph with the only exception that G is a 4-regular graph with λ3(G) = 4. Furthermore, λ3(G) = 4 if and only if G contains K4 as its subgraph.NationalNaturalScienceFoundationofChina(10271105) FoundationofEducationMinistryofFujian(JA03147) MinistryofScienceandTechnologyofFujian(2003J036