The number of vertices of degree 7 in a contraction-critical 7-connected graph

Abstract

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 V 7 the set of vertices of degree 7 in G . In this paper, we prove that | V 7 | ≥ n 22 , which improves the result proved by Ando, Kaneko and Kawarabayashi. In the meantime, we obtain that for any vertex x ⁄ ∈ V 7 in a contraction-critical 7-connected graph there is a vertex y ∈ V 7 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 V 7 is independent.