Trivially noncontractible edges in a contraction critically 5-connected graph

Abstract

An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. A k-connected graph with no k-contractible edge is said to be contraction critically k-connected. An edge of a k-connected graph is said to be trivially noncontractible if its end vertices have a common neighbor of degree k. We prove that a contraction critically 5-connected graph on n vertices has at least n / 2 trivially noncontractible edges and at least ( 2 n ) / 9 vertices of degree 5.