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 χa(G) and χs(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 χa(G′), χs(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.