author: | David R. Wood |
---|---|

title: | Acyclic, Star and Oriented Colourings of Graph Subdivisions |

keywords: | graph, graph colouring, star colouring, star chromatic number, acyclic colouring, acyclic chromatic number, oriented colouring, oriented chromatic number, subdivision |

abstract: | 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 and
_{a}(G)χ denote the acyclic and star
chromatic numbers of _{s}(G)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
and _{s}(G')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 of an
(undirected) graph ^{→}(G)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
whenever ^{→}(G')=χ(G)χ(G)≥9.
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reference: | David R. Wood (2005),
Acyclic, Star and Oriented Colourings of Graph Subdivisions,
Discrete Mathematics and Theoretical Computer Science 7, pp. 37-50 |

bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |

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