The number of vertices of degree 5 in a contraction-critically 5-connected graph

An edge of a 5-connected graph is said to be 5-contractible if the contraction of the edge results in a 5-connected graph. A 5-connected graph with no 5-contractible edge is said to be contraction-critically 5-connected. Let V(G)V(G) and V5(G)V5(G) denote the vertex set of a graph GG and the set of degree 5 vertices of GG, respectively. We prove that each contraction-critically 5-connected graph GG has at least |V(G)|/2|V(G)|/2 vertices of degree 5. We also show that there is a sequence of contraction-critically 5-connected graphs {Gi}{Gi} such that limi→∞|V5(Gi)|/|V(Gi)|=1/2limi→∞|V5(Gi)|/|V(Gi)|=1/2.