## Discrete Mathematics & Theoretical Computer Science, Vol 7 (2005)

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DMTCS vol 7 no 1 (2005), pp. 37-50

# Discrete Mathematics & Theoretical Computer Science

## Volume 7 n° 1 (2005), pp. 37-50

author: David R. Wood Acyclic, Star and Oriented Colourings of Graph Subdivisions graph, graph colouring, star colouring, star chromatic number, acyclic colouring, acyclic chromatic number, oriented colouring, oriented chromatic number, subdivision 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. If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files. David R. Wood (2005), Acyclic, Star and Oriented Colourings of Graph Subdivisions, Discrete Mathematics and Theoretical Computer Science 7, pp. 37-50 For a corresponding BibTeX entry, please consider our BibTeX-file. dm070104.ps.gz (59 K) dm070104.ps (168 K) dm070104.pdf (123 K)