Extremal results regarding $K_6$-minors in graphs of girth at least 5

Abstract

We prove that every 6-connected graph of girth $\ge 6$ has a $K_6$-minor and thus settle the Jorgensen conjecture for graphs of girth $\ge 6$. Relaxing the assumption on the girth, we prove that every 6-connected $n$-vertex graph of size $\ge 31/5n-8$ and of girth $\ge 5$ contains a $K_6$-minor.

13 pages, submitted on October 12 2010