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- W4315589038 abstract "Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for digraphs with maximum degree $5$ or oriented planar graphs with maximum degree $6$. A digraph is said to be $k$-mixing if there exists a transformation between any pair of $k$-colourings. We show that every digraph $D$ is $k$-mixing for all $kgeq delta^*_{min}(D)+2$, generalizing a result due to Dyer et al. We also prove that every oriented graph $vec{G}$ is $k$-mixing for all $kgeq delta^*_{max}(vec{G}) +1$ and for all $kgeq delta^*_{rm avg}(vec{G})+1$. We conjecture that, for every digraph $D$, the dicolouring graph of $D$ on $kgeq delta_{min}^*(D)+2$ colours has diameter at most $O(|V(D)|^2)$ and give some evidences. We first prove that the dicolouring graph of any digraph $D$ on $kgeq 2delta_{min}^*(D) + 2$ colours has linear diameter, extending a result from Bousquet and Perarnau. We also prove that the conjecture is true when $kgeq frac{3}{2}(delta_{min}^*(D)+1)$. Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on $kgeq 2$ colours is connected and has diameter at most $2n$. We conjecture that every non $2$-mixing oriented graph has maximum average degree at least $4$, and we provide some support for this conjecture by proving it on the special case of $2$-freezable oriented graphs. More generally, we show that every $k$-freezable oriented graph on $n$ vertices must contain at least $kn + k(k-2)$ arcs, and we give a family of $k$-freezable oriented graphs that reach this bound. In the general case, we prove as a partial result that every non $2$-mixing oriented graph has maximum average degree at least $frac{7}{2}$." @default.
- W4315589038 created "2023-01-11" @default.
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- W4315589038 date "2023-01-09" @default.
- W4315589038 modified "2023-10-06" @default.
- W4315589038 title "Digraph redicolouring" @default.
- W4315589038 doi "https://doi.org/10.48550/arxiv.2301.03417" @default.
- W4315589038 hasPublicationYear "2023" @default.
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