Math Dept Research Series: The Extremal Function K10 for Minors

 Abstract: 

 For positive integers $t$ and $n$, the maximum number of edges that an $n$-vertex graph with no $K_t$ minor can have is known as the extremal function for $K_t$ minors. In 1968, Mader proved that for every integer $t = 1, 2, ..., 7$, a graph on $n \geq t$ vertices and at least $(t-2)n - \binom{t-1}{2} + 1$ edges has a $K_t$ minor. J{\o}rgensen showed that a graph on $n \geq 8$ vertices and at least $6n - 20$ edges either has a $K_8$ minor or is isomorphic to a graph obtained from disjoint copies of $K_{2, 2, 2, 2, 2}$ by identifying cliques of size 5. Song and Thomas further generalized the results for $K_9$ minors. These known extremal functions for $K_t$ minors have been crucial for proving several results related to Hadwiger's conjecture for small clique minors, and the Hadwiger's conjecture is a generalization of the famous Four Color Theorem. In this talk, I will discuss my work on the extremal function for $K_{10}$ minors. This is the core of my PhD thesis at Georgia Tech, mentored by both Robin Thomas and Xingxing Yu.