Acyclic, Star and Oriented Colourings of Graph Subdivisions

Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χ a (G) and χ s (G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G' be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G' correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χ a (G'), χ s (G') and χ(G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The oriented chromatic number χ → (G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that χ → (G')=χ(G) whenever χ(G)≥9.