#### Date of Award

1-1-2022

#### Document Type

Dissertation

#### Degree Name

Ph.D. in Mathematics

#### Department

Mathematics

#### First Advisor

Bing Wei

#### Second Advisor

Haidong Wu

#### Third Advisor

Talmage James Reid

#### Relational Format

dissertation/thesis

#### Abstract

In 1930, Frank Ramsey showed that one will find a monochromatic clique of a specified size in any edge coloring of a sufficiently large complete graph. This result is commonly known as Ramsey's theorem. The role of Ramsey number is to quantify some specific problems related to Ramsey's theorem. Given a graph H, the Ramsey number R_2(H) is the minimum integer p such that every 2-edge-coloring of the complete graph K_p contains a monochromatic copy of H. Ramsey numbers have been a hot topic in graph theory for decades due to their intrinsic beauty, wide applicability and overwhelming difficulty. Let F_n be a graph consisting of n triangles, all sharing one common vertex, and S_{t,r} (r\ge2) be a graph obtained from a star with t vertices by adding a path P_r which connects r distinct leaves of the star. Chen, Yu and Zhao (2021) speculated that R_2(F_n)\le R_2(nK_3)=5n for some n\ge2. However, the exact value for R_2(F_n) is very hard to determine. So far, it has been verified for n=2. We confirm their assertion for n=3 by proving that R_2(F_3)=14. We also study the Ramsey numbers of S_{t,r} and obtain the exact values of R_2(S_{t,r}) for 2\le r\le4.

#### Recommended Citation

Zhao, Qinghong, "On Ramsey and Gallai-Ramsey numbers of some classes of graphs" (2022). *Electronic Theses and Dissertations*. 2299.

https://egrove.olemiss.edu/etd/2299