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Discussiones Mathematicae Graph Theory 19(2) (1999) 175-197
DOI: https://doi.org/10.7151/dmgt.1094

ON-LINE RANKING NUMBER FOR CYCLES AND PATHS

Erik Bruoth and Mirko Hornák

Department of Geometry and Algebra
P.J. Safárik University, Jesenná 5
041 54 Košice, Slovakia

e-mail: ebruoth@duro.upjs.sk
e-mail: hornak@turing.upjs.sk

Abstract

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^*(P_n) < 3log₂n for n ≥ 2. Here we show that χ_r^*(P_n) ≤ 2⎣log₂n⎦+1. The same upper bound is obtained for χ_r^*(C_n),n ≥ 3.

Keywords: ranking number, on-line vertex colouring, cycle, path.

1991 Mathematics Subject Classification: 05C15.

References

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Received 22 February 1999
Revised 29 October 1999