#### 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