Suppose we generalize the “cut rule” (in the implementation of decrease-key operation for a Fibonacci heap) to cut a node x from its parent as soon as it loses its kth child, for some integer constant...

Suppose we generalize the “cut rule” (in the implementation of decrease-key operation for a Fibonacci heap) to cut a node x from its parent as soon as it loses its kth child, for some integer constant k. (The rule that we studied uses k = 2.) For what values of k can we upper bound the maximum degree of a node of an n-node Fibonacci heap with O(log n)?