**Title:** 6-connected graphs are two-three linked

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

**Committee:**

Dr. Robin Thomas, School of Mathematics, Georgia Institute of Technology

Dr. Prasad Tetali, School of Mathematics, Georgia Institute of Technology

Dr. Lutz Warnke, School of Mathematics, Georgia Institute of Technology

Dr. Richard Peng, School of Computer Science, Georgia Institute of Technology

**Reader:** Dr. Gexin Yu, Department of Mathematics, College of William and Mary

**Summary:** Let $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on J\o rgensen's conjecture, Robertson and Seymour asked when does $G$ contain disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We prove that if $G$ is 6-connected then such $G_1,G_2$ exist. Joint work with Robin Thomas and Xingxing Yu.