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

Kriesell has proved that every noncomplete κ -connected ( κ ≠ 7 ) graph has a κ -contractible edge (i.e., an edge whose contraction yields again a κ -connected graph), if the degree sum of any two vertices with distance at most two is at least 2 ⌊ 5 κ 4 ⌋ - 1 . And he guessed that the degree sum condition can be relaxed for only adjacent pairs. In this paper, we verified that is true for κ ⩾ 8 .