Let be a digraph. We define as the maximum of and as the maximum of . It is known that the dichromatic number of is at most . In this work, we prove that every digraph which has dichromatic number exactly must contain the directed join of and for some such that , except if in which case must contain a digon. In particular, every oriented graph with has dichromatic number at most . Let be an oriented graph of order such that . Given two 2-dicolourings of , we show that we can transform one into the other in at most steps, by recolouring exactly one vertex at each step while maintaining a dicolouring at any step. Furthermore, we prove that, for every oriented graph on vertices, the distance between two -dicolourings is at most when . We then extend a theorem of Feghali, Johnson and Paulusma to digraphs. We prove that, for every digraph with and every , the -dicolouring graph of consists of isolated vertices and at most one further component that has diameter at most , where is a constant depending only on .

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强化有向图的有向布鲁克斯定理以及有向图重着色的后果

让是一个有向图。我们定义作为最大值和作为最大值。已知二色数为至多是。在这项工作中，我们证明每个有向图正好有二色数必须包含定向连接和对于一些这样，除非如果在这种情况下必须包含一个二边形。特别是，每个有向图和最多有二色数。让是一个有序的有向图这样。给定两个 2-双色，我们证明我们最多可以将一种转变为另一种步骤，通过在每个步骤中精确地重新着色一个顶点，同时在任何步骤中保持双色。此外，我们证明，对于每个有向图在顶点，两点之间的距离-双色最多什么时候。然后我们将 Feghali、Johnson 和 Paulusma 的定理推广到有向图。我们证明，对于每个有向图和和每一个， 这-双色图由孤立的顶点和至多一个直径至多的其他组件组成， 在哪里是一个常数，仅取决于。