有向图重染色
Digraph redicolouring
AI总结：
本文研究有向图重染色的 PSPACE 完全性、k-混合阈值与染色图直径，并给出定向图最大平均度下界及可冻结图的紧弧数界。
摘要：
给定有向图 D 的两个 k-有向染色，我们证明：即使在 k=2 且有向图的最大度为 5，或最大度为 6 的定向平面图的情况下，判定我们能否通过每一步只重染一个顶点、并在任意步骤都保持有向染色的方式将一个染色转化为另一个染色，仍是 PSPACE 完全的。如果任意一对 k-染色之间都存在一种转化，则称该有向图是 k-混合的。我们证明，每个有向图 D 对所有 k≥δ*_{min}(D)+2 都是 k-混合的，这推广了 Dyer 等人的结果。我们还证明，每个定向图 \vec{G} 对所有 k≥δ*_{max}(\vec{G})+1 以及所有 k≥δ*_{avg}(\vec{G})+1 都是 k-混合的。我们猜想，对每个有向图 D，其在 k≥δ*_{min}(D)+2 种颜色上的有向染色图直径至多为 O(|V(D)|^2)，并给出了一些证据。我们首先证明，任意有向图 D 在 k≥2δ*_{min}(D)+2 种颜色上的有向染色图具有线性直径，这扩展了 Bousquet 和 Perarnau 的结果。我们还证明当 k≥\frac{3}{2}(δ*_{min}(D)+1) 时该猜想成立。限于定向图这一特殊情形，我们证明任意次立方定向图在 k≥2 种颜色上的有向染色图是连通的，且直径至多为 2n。我们猜想每个非 2-混合定向图的最大平均度至少为 4，并通过在 2-可冻结定向图这一特殊情形下证明该猜想，为其提供了支持。更一般地，我们表明每个含 n 个顶点的 k-可冻结定向图必须至少包含 kn+k(k−2) 条弧，并给出了一个达到该界的 k-可冻结定向图族。在一般情况下，作为部分结果，我们证明每个非 2-混合定向图的最大平均度至少为 \frac{7}{2}。
英文摘要：
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 $k\geq δ^*_{\min}(D)+2$, generalizing a result due to Dyer et al. We also prove that every oriented graph $\vec{G}$ is $k$-mixing for all $k\geq δ^*_{\max}(\vec{G}) +1$ and for all $k\geq δ^*_{\rm avg}(\vec{G})+1$. We conjecture that, for every digraph $D$, the dicolouring graph of $D$ on $k\geq δ_{\min}^*(D)+2$ 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 $k\geq 2δ_{\min}^*(D) + 2$ colours has linear diameter, extending a result from Bousquet and Perarnau. We also prove that the conjecture is true when $k\geq \frac{3}{2}(δ_{\min}^*(D)+1)$. Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on $k\geq 2$ 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 $\frac{7}{2}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
