Final doctoral examination and defense of dissertation of Yingjie Qian, June 23, 2022
Thesis Title: Non-separating paths in graphs
Link to thesis draft: https://drive.google.com/file/d/1_8rVdmLHm_gSGP8r9HnXdP33ASC4XJy1/view?u...
Date: June 23, 2022
Time: 10AM (EDT)
Location (in-person): Skiles 006
Link (virtual): https://gatech.zoom.us/j/97675110493?pwd=YThKMFpBMnd2VE52Rmx3NWxocEZDUT09
Meeting ID: 976 7511 0493 Passcode: acograph
Committee:
Dr. Anton Bernshteyn, School of Mathematics, Georgia Institute of Technology
Dr. Grigoriy Blekherman, School of Mathematics, Georgia Institute of Technology
Dr. Zi-Xia Song, Department of Mathematics, University of Central Florida
Dr. Zhiyu Wang, School of Mathematics, Georgia Institute of Technology
Dr. Xingxing Yu, School of Mathematics, Georgia Institute of Technology
Advisor: Dr. Xingxing Yu, School of Mathematics, Georgia Institute of Technology
Reader: Dr. Zi-Xia Song, Department of Mathematics, University of Central Florida
Abstract:
When developing a theory for 3-connected graphs, Tutte showed that for any 3-connected graph G and any three vertices a, b, c of G, G-c has an a-b path P such that G-P is connected. We call paths non-separating if their removal results in a graph satisfying a certain connectivity constraint. There is a series of work on non-separating paths in graphs and their applications.
For any graph G and distinct vertices a,b,c,d in V(G), we give a structural characterization for G not containing a path A from a to b and avoiding c and d such that removing A from G results in a 2-connected graph. Using this structure theorem, we construct a 7-connected such graph. We will also discuss potential applications to other problems, including the 3-linkage conjecture made by Thomassen in 1980.
