A Gray code for arborescences of tournaments

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 $G$ 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.