Some structural properties of minimally contraction-critically 5-connected graphs

An edge of a k -connected graph is said to be k -removable (resp. k -contractible) if the removal (resp. the contraction ) of the edge results in a k -connected graph. A k -connected graph with neither k -removable edge nor k -contractible edge is said to be minimally contraction-critically k -connected. We show that around an edge whose both end vertices have degree greater than 5 of a minimally contraction-critically 5-connected graph, there exists one of two specified configurations. Using this fact, we prove that each minimally contraction-critically 5-connected graph on n vertices has at least 2 3 n vertices of degree 5.