Final doctoral examination and defense of dissertation of Dantong Zhu, October 15, 2021

Date: October 15, 2021, 1:00pm EST

Bluejeans Information:

Meeting URL: https://bluejeans.com/961599338/0497?src=join_info

Meeting ID: 961 599 338

Participant Passcode: 0497

Title: The Extremal Function for K10 Minors

Advisor: Dr. Xingxing Yu, School of Mathematics, Georgia Institute of Technology

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

Reader: Dr. Zi-Xia Song, Department of Mathematics, University of Central Florida

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Link to thesis draft is available here:

https://drive.google.com/file/d/1SFBL9sdfOtvpiGwh6WhJPA1NZVHnazpo/view?u...

Summary of the thesis is below.

Summary:

For two graphs G and H, G has H as a minor if a graph isomorphic to H can be obtained from a subgraph of G by repeatedly contracting edges. The Four Color Theorem (FTC) says that every planar graph is 4-colorable. Due to the Kuratowski-Wagner theorem, the FTC can be restated that every graph with no K5 minor and K3,3 minor is 4-colorable. The famous Hadwiger Conjecture is a generalization of the FTC, which says that every graph with no K_{t+1} minor for integers t >= 1 is t-colorable. The Hadwiger's Conjecture is true for all t = 6.

To make progress on Hadwiger's Conjecture for t >= 6, one major line of work has focused on giving an upper bound on the number of edges for graphs without a Kt minor. The maximum number of edges of an n-vertex graph with no Kt minor is known as the extremal function for Kt minors. This dissertation focuses on the extremal function for K10 minors. We prove that every graph on n >= 8 vertices and at least 8n - 35 edges either has a K10 minor or falls into a few families of exceptional graphs.