Redicolouring digraphs: directed treewidth and cycle-degeneracy

TL;DR

This work extends the theory of graph recolouring to digraphs by introducing cycle-degeneracy $\delta^*_c(D)$ as a directed analogue of undirected degeneracy and studying the diameter and mixing properties of the digraph dicolouring graph $\mathcal{D}_k(D)$. It establishes tight-to-near-tight bounds across multiple regimes: for $k \ge \delta^*_c(D)+2$ and related thresholds, and for crosses with maximum average cycle-degree $\mathrm{mad}_c(D)$ and digrundy-related parameters, linking to directed treewidth via $\mathscr{D}$-width and to the underlying undirected graph. The results unify several undirected recolouring techniques with directed analogues, provide transfer principles through the underlying graph, and extend planar and other special-case recolouring analyses to planar digraphs, offering new tools for directed reconfiguration problems. Overall, the paper advances understanding of when and how digraphs can be recoloured efficiently, with implications for directed width measures and planarity-driven reconfiguration questions.

Abstract

Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of cycle-degree, we define the $c$-degeneracy (or cycle-degeneracy) of $D$, which we denote by $δ^*_c(D)$. It appears to be a nice generalisation of the undirected degeneracy. In this work, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda's conjecture to digraphs. The $k$-dicolouring graph of $D$, denoted by $\mathcal{D}_k(D)$, is the undirected graph whose vertices are the $k$-dicolourings of $D$ and in which two $k$-dicolourings are adjacent if they differ on the colour of exactly one vertex. We show that $\mathcal{D}_k(D)$ has diameter at most $O_{δ^*_c(D)}(n^{δ^*_c(D) + 1})$ (respectively $O(n^2)$ and $(δ^*_c(D)+1)n$) when $k$ is at least $δ^*_c(D)+2$ (respectively $\frac{3}{2}(δ^*_c(D)+1)$ and $2(δ^*_c(D)+1)$). This improves known results on digraph redicolouring (Bousquet et al.). Next, we extend a result due to Feghali to digraphs, showing that $\mathcal{D}_{d+1}(D)$ has diameter at most $O_{d,ε}(n(\log n)^{d-1})$ when $D$ has maximum average cycle-degree at most $d-ε$. We then show that two proofs of Bonamy and Bousquet for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph and the second one uses the $\mathscr{D}$-width. Finally, we give a general theorem which makes a connection between the recolourability of a digraph $D$ and the recolourability of its underlying graph $UG(D)$. This result directly extends a number of results on planar graph recolouring to planar digraph redicolouring.

Figures (2)

• Figure 1: The redicolouring sequence between $\gamma_i$ and $\gamma_{i+\left \lceil \frac{k}{3} \right \rceil}$.
• Figure 2: The structure of the digraph $D$ dicoloured with $\alpha$, which we assume to be an ($s+2$)-dicolouring for clarity. Note that $W_c$ does not exist and $X_u$ may intersect $V_{q+1}\cup \dots \cup V_{h-1}$.

Theorems & Definitions (69)

• Theorem 1: Bonsma and Cereceda bonsmaTCS410 ; Dyer et al. dyerRSA29
• Conjecture 2: Cereceda cerecedaTHESIS
• Theorem 3: Bousquet and Heinrich bousquetJCTB155
• Theorem 4: Bousquet and Perarnau bousquetEJC52
• Theorem 5: Feghali feghaliJCT147
• Theorem 6: Bonamy and Bousquet bonamyEJC69
• Theorem 7: Bonamy and Bousquet bonamyEJC69
• Theorem 8: Bousquet et al. bousquetArXiv23
• Conjecture 9: Bousquet et al. bousquetArXiv23
• Theorem 10: Picasarri-Arrieta picasarriArxiv23