Geodesic
Detour chromatic numbers
Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 2, pp. 283-291

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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 
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Frick, Marietjie; Bullock, Frank. Detour chromatic numbers. Discussiones Mathematicae. Graph Theory, Tome 21 (2001) no. 2, pp. 283-291. http://geodesic.mathdoc.fr/item/DMGT_2001_21_2_a11/