A degree sum condition for the existence of a contractible edge in a k-connected graph

It is known that a noncomplete kappa -connected graph of minimum degree of at least right perpendicular 5 kappa /4 left perpendicular contains a kappa -contractible edge, i.e., an edge whose contraction yields again a kappa - connected graph. Here we prove the stronger statement that a noncomplete ka ppa -connected graph for which the sum of the degrees of any two distinct v ertices is at least 2 right perpendicular 5/4 kappa left perpendicular - 1 possesses a kappa -contractible edge. The bound is sharp and remains valid and sharp if we look only at degree sums at pairs of vertices at distances of one or two, provided that kappa not equal 7. (C) 2001 Academic Press.