A Degree Sum Condition for the Existence of a Contractible Edge in a κ-Connected Graph

It is known that a noncomplete κ-connected graph of minimum degree of at least ⌊5κ4⌋ contains a κ-contractible edge, i.e., an edge whose contraction yields again a κ-connected graph. Here we prove the stronger statement that a noncomplete κ-connected graph for which the sum of the degrees of any two distinct vertices is at least 2⌊54 κ⌋−1 possesses a κ-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 κ≠7.