# Highly connected graphs with no K_n minor

**Problem**Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?

A famous conjecture of Jorgensen asserts that every 6-connected graph without a -minor is apex (planar plus one vertex). If true, Jorgensen's conjecture does not generalize (naively) to higher connectivities, since for sufficiently large , there do exist -connected graphs which are not close to planar in the sense we are considering (many more than vertices must be deleted to leave a planar graph). This conjecture of Thomas asserts that all such graphs are small in size.

For this conjecture is true. For this conjecture is trivial, since any graph without a -minor is planar. The case follows from a theorem of Wagner which gives a construction for all graphs without -minors (and from which it follows that every 4-connected graph with no minor is planar). The case was recently resolved by DeVos, Hegde, Kawarabayashi, Norine, Thomas, and Wollan. The difficulties associated with finding minors in graphs make this conjecture appear daunting, but if true, it would yield powerful insight into the structure of graphs.