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

TL;DR

This paper proves that highly connected graphs with large girth contain a complete minor of size six, settling a conjecture for girth at least six and extending results to girth five with size conditions.

Contribution

It establishes the existence of a $K_6$-minor in 6-connected graphs with girth at least 6 and extends to graphs with girth 5 under size constraints, settling the Jorgensen conjecture in these cases.

Findings

6-connected girth ≥ 6 graphs contain a $K_6$-minor

Graphs with girth ≥ 5 and sufficient size contain a $K_6$-minor

Settles the Jorgensen conjecture for girth ≥ 6

Abstract

We prove that every 6-connected graph of girth $\geq 6$ has a $K_6$-minor and thus settle the Jorgensen conjecture for graphs of girth $ \geq 6$. Relaxing the assumption on the girth, we prove that every 6-connected $n$-vertex graph of size $\geq 3 1/5 n-8$ and of girth $\geq 5$ contains a $K_6$-minor.