Contractible edges in k-connected graphs containing no K4−

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. Let K4− stand for the graph obtained from K4 by removing one edge. It is proved by the third author of this paper that if k is odd, then a k-connected graph which does not contain K4− has a k-contractible edge. The same conclusion does not hold when k is even. In this paper, we prove that if a k-connected graph with no K4− does not have a k-contractible edge, then k is even and each vertex in G is contained in at least two triangles.