Acyclic Dicolourings of Oriented Graphs: Paths, Random Tournaments, and Critical Orders

Abstract

An acyclic dicolouring of an oriented graph is a vertex partition in which every colour class and every bipartite subdigraph induced by two classes is acyclic. We first prove the Gallai--Roy-type bound $\vecχ_a(D)\leq L(D)$, where $L(D)$ is the maximum order of a directed path. Let $r=\log_2(8/7)$. Bang-Jensen, Picasarri-Arrieta, and Yeo previously constructed tournaments of order $n$ whose acyclic dichromatic number is at least $n-(\frac8r)\log_2 n-\log_2\log_2 n$. We improve the leading logarithmic coefficient by a factor of two: for the uniform random tournament $\mathcal{T}_n$, asymptotically almost surely, $\vecχ_a(\mathcal{T}_n)\geq n-\frac4r\log_2 n+\frac2r\log_2\log_2 n-O(1)$. Finally, if $m(k)$ denotes the minimum order of an oriented graph with acyclic dichromatic number at least $k$, tournament completion shows that the same minimum is obtained over tournaments. We prove $m(3)=5$ and $m(4)=7$ and classify the tournament witnesses at these minimum orders: there is one at order five and two at order seven.