A Gray code for arborescences of tournaments

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.

Figures (14)

• Figure 1: Two Hamiltonian paths in a ladder with extremities at different levels. The paths $P_1$ and $P_3$ are drawn in blue, the path $P_2$ in red.
• Figure 2: A directed graph $G$ for which contracting $xz$ into $u$ creates a flip unfeasible in $G$. Throughout this article, we will consistently represent the root of our directed graphs by a circled vertex, here $r$.
• Figure 3: Constructing a Hamiltonian path of $\mathcal{F}_{r}(G')$ from a Hamiltonian path of $\mathcal{F}_{r}(G)$. In the Hamiltonian path of $\mathcal{F}_{r}(G)$ on the left, the vertices that correspond to arborescences containing $e$ are drawn in red. On the right a Hamiltoninan path of $\mathcal{F}_{r}(G')$. The vertices corresponding to arborescences containing $e$ are drawn in red, those containing $e'$ in blue. The dotted edges are edges that are present in the flip graph, but not used in the Hamiltonian path.
• Figure 4: The flip graph of a bidirected 5-cycle
• Figure 5: A graph with 13 arborescences and no Hamiltonian cycle in its flip graph. On the right, its weighted Laplace matrix (defined in \ref{['thm:balanced_bipartition']}), where the $i$-th row and column correspond to the vertex labelled $i$.

Theorems & Definitions (30)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Lemma 6
• proof
• Lemma 7
• proof