Minimally contraction-critically 6-connected graphs

An edge of a 6 -connected graph is said to be removable (resp. contractible) if the removal (resp. contraction) of the edge results in a 6 -connected graph. A 6 -connected graph is said to be minimally contraction-critically 6 -connected if it has neither removable edge nor contractible edge. Let x be a vertex of a minimally contraction-critically 6 -connected graph G . In this paper, we show that there is one of some specified configurations around x and using this result we prove that x has a neighbor of degree 6. We also display a condition for x to have at least two neighbors of degree 6.