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 ) and V 5 ( G ) denote the vertex set of a graph G and the set of degree 5 vertices of G , respectively. We prove that each contraction-critically 5-connected graph G has at least | V ( G ) | / 2 vertices of degree 5. We also show that there is a sequence of contraction-critically 5-connected graphs { G i } such that lim i → ∞ | V 5 ( G i ) | / | V ( G i ) | = 1 / 2 .