Date of Award
1-1-2019
Document Type
Dissertation
Degree Name
Ph.D. in Mathematics
First Advisor
Haidong Wu
Second Advisor
Qingying Bu
School
University of Mississippi
Relational Format
dissertation/thesis
Abstract
The beta invariant of a matroid was introduced by Crapo in 1967. We first find the lower bound of the beta invariant of 3-connected matroids with rank r and the matroids which attain the lower bound. Second we characterize the matroids with beta invariant 5 and 6. For binary matroids we characterize matroids with beta invariant 7. These results extend earlier work of Oxley. Lastly we partially answer an open question of chromatic uniqueness of wheels and prove a splitting formula for the beta invariant of generalized parallel connection of two matroids. Tutte's Wheel-and-Whirl theorem and Seymour's Splitter theorem give respectively a constructive and structural view of the 3-connected matroids. Geelen and Whittle proved a chain theorem for sequentially 4-connected matroids and Geelen and Zhou proved a chain theorem for weakly 4-connected matroids. From these theorems one can obtain a chain theorem for matroids as well. We prove a chain theorem for sequentially 4-connected and weakly 4-connected matroids.
Recommended Citation
Lee, Sooyeon, "Beta invariant and variations of chain theorems for matroids" (2019). Electronic Theses and Dissertations. 1772.
https://egrove.olemiss.edu/etd/1772