|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
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
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
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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.|
|ps.gz-source:||dm070104.ps.gz (59 K)|
|ps-source:||dm070104.ps (168 K)|
|pdf-source:||dm070104.pdf (123 K)|
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