A Gray code for arborescences of tournaments
Marthe Bonamy, Michael Hoffmann, Cl\'ement Legrand-Duchesne, G\"unter Rote

TL;DR
This paper investigates the existence of Gray codes for arborescences in tournaments, showing positive results for certain cases and analyzing conditions where Hamiltonian cycles in flip graphs may not exist.
Contribution
It provides a construction of Gray codes for arborescences in tournaments and examines conditions affecting Hamiltonian cycles in flip graphs.
Findings
Positive answer for Gray codes in tournaments
Conditions where flip graphs lack Hamiltonian cycles
Analysis of reconfiguration graphs for arborescences
Abstract
We consider the following question of Knuth: given a directed graph and a root , can the arborescences of rooted in 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 under arc flips, also called flip graph of . 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.
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