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

An edge of a k-connected graph is said to be k-contractible if its contraction yields a k-connected graph. A non-complete k-connected graph possessing no k-contractible edges is called contraction-critical k-connected. Let G be a contraction-critical 7-connected graph with n vertices, and V7 the set of vertices of degree 7 in G. In this paper, we prove that |V7| ≥ n 22 , which improves the result proved by Ando, Kaneko and Kawarabayashi. In the meantime, we obtain that for any vertex x 6∈ V7 in a contraction-critical 7-connected graph there is a vertex y ∈ V7 such that the distance between x and y is at most 2, and thus extends a result of Su and Yuan. We present a family of contraction-critical 7-connected graphs G in which V7 is independent. c © 2007 Elsevier B.V. All rights reserved.