**Conjecture**Every graph with minimum degree at least 7 contains a -minor.

**Conjecture**Every 7-connected graph contains a -minor.

The first conjecture implies the second.

Whether the second conjecture is true was first asked in [KT]. Both conjectures were stated in [BJW].

The second conjecture is implied by Jørgensen’s conjecture, which asserts that every -connected -minor-free graph is apex (which have minimum degree at most and are thus not -connected). Since Jørgensen’s conjecture is true for sufficiently large graphs [KNTWa,KNTWb], the second conjecture is true for sufficiently large graphs.

## Bibliography

*[BJW] János Barát, Gwenaël Joret, David R. Wood. Disproof of the List Hadwiger Conjecture, Electronic J. Combinatorics 18:P232, 2011.

*[KT] Ken-ichi Kawarabayashi and Bjarne Toft. Any 7-chromatic graph has or as a minor. Combinatorica 25 (3), 327–353, 2005.

[KNTWa] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. minors in -connected graphs of bounded tree-width. http://arxiv.org/abs/1203.2171

[KNTWb] Ken-ichi Kawarabayashi, Serguei Norine, Robin Thomas, Paul Wollan. minors in large -connected graphs. http://arxiv.org/abs/1203.2192

* indicates original appearance(s) of problem.