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 χr*(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 χr*(Pn) < 3log2n for n ≥ 2. Here we show that χr*(Pn) ≤ 2⎣log2n⎦+1. The same upper bound is obtained for χr*(Cn),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
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