ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 21(1) (2001) 283-291
DOI: 10.7151/dmgt.1150


Marietjie Frick and Frank Bullock

University of South Africa
P.O. Box 392, Unisa, 0003
South Africa


The nth detour chromatic number, χn(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ( G). We conjecture that χn(G) ≤ ⎡[(τ(G))/n]⎤ for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some sufficient conditions for a graph to have nth chromatic number at most 2.

Keywords: detour, generalised chromatic number, longest path, vertex partition, girth, circumference, nearly bipartite.

2000 Mathematics Subject Classification: 05C15, 05C38.


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Received 19 February 2001
Revised 8 October 2001