A Gray code for arborescences of tournaments

2026-03-30

AI Summary

This study addresses the reconfiguration problem of arborescences rooted at a specified vertex in tournament graphs, aiming to construct a pivot Gray code—an enumeration sequence in which consecutive arborescences differ by exactly one arc. By modeling the problem via a flip graph and analyzing its Hamiltonian properties, the authors establish for the first time the existence of such a Gray code for arborescences in tournaments. Additionally, they identify several sufficient conditions under which the flip graph of arborescences in a general directed graph fails to contain a Hamiltonian cycle. Bridging graph-theoretic reconfiguration, flip graph modeling, and Hamiltonicity analysis, this work provides both theoretical foundations and novel insights for the efficient enumeration of combinatorial structures.

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.

Problem

• Gray code
• arborescence
• tournament
• Hamiltonian path
• flip graph

Innovation

• pivot Gray code
• arborescence
• tournament
• flip graph
• Hamiltonian path