Speaker：Rong Luo, Department of Mathematics
West Virginia University Morgantown
Time: 20170607 10:0011:00
Place：Room 1518，School of Mathematical
Sciences
Detail： The concept of group connectivity was
introduced by Jaeger, Linial, Payan, and Tarsi (Journal Combinatorial Theory,
Ser. B, 1992) as a generalization of nowherezero group flows. Let A be an
Abelian group. An Aconnected graphs are contractible configurations of Aflow
and play an important role in the study of group flows because of the fact: if
H is Aconnected, then any supergraph G of H (i.e. G contains H as a subgraph)
admits a nowherezero Aflow if and only if G/H does. It is known that an Aconnected
graph cannot be very sparse. How dense could an Aconnected graph be? This
motivates us to study the extremal problem: find the maximum integer k, denoted
ex(n, A), such that every graph with at most k edges is not Aconnected. We
determine the exact values for all finite cyclic groups. As a corollary, we
present a characterization of all Zkconnected graphic sequences. As noted by
Jaeger, Linial, Payan, and Tarsi, there are Z5connected graph that are not
Z6connected. We also prove that every Z3connected graph contains two
edgedisjoint spanning trees, which implies that every Z3connected graph is
also Aconnected for any Abelian group A with order at least 4. In the second
part of the talk, I will introduce the concept of group connectivity of signed
graphs and present some basic properties of group connectivities of signed
graphs.
Organizer:
School of Mathematical Sciences
