Discrete Mathematics & Theoretical Computer Science
Volume 7 n° 1 (2005), pp. 37-50
| 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
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.
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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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