Geodesic
On-line ranking number for cycles and paths
Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 2, pp. 175-197

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A k-ranking of a graph G is a colouring φ:V(G) → 1,...,k such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number χ*_r(G) of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that χ*_r(Pₙ) 3log₂n for n ≥ 2. Here we show that χ*_r(Pₙ) ≤ 2⎣log₂n⎦+1. The same upper bound is obtained for χ*_r(Cₙ),n ≥ 3.
Keywords: ranking number, on-line vertex colouring, cycle, path 
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     title = {On-line ranking number for cycles and paths},
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Bruoth, Erik; Horňák, Mirko. On-line ranking number for cycles and paths. Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 2, pp. 175-197. http://geodesic.mathdoc.fr/item/DMGT_1999_19_2_a5/