We consider the following question of Knuth: given a directed graph G and a root r, can the arborescences of G rooted in r be listed such that any two consecutive arborescences differ by only one arc? Such an ordering is called a pivot Gray code and can be formulated as a Hamiltonian path in the reconfiguration graph of the arborescences of G under arc flips, also called flip graph of G. We give a positive answer for tournaments and explore several conditions showing that the flip graph of a directed graph may contain no Hamiltonian cycles.
Last update: April 30, 2026.