A lower bound for on-line ranking number of a path

A k-ranking of a graph G is a mapping phi : V (G) -> {1,...,k} such that any path with endvertices x and y satisfying x not equal y and phi(x) = phi(y) contains a vertex z with phi(z) > phi(x). The ranking number chi(*)(r)(G) of G is the minimum k admitting a k-ranking of G. The on-line ranking number chi(*)(r) (G) of G is the corresponding on-line invariant; in that case vertices of G are coming one by one so that a partial ranking has to be chosen by considering only the structure of the subgraph of G induced by the present vertices. It is known that [log(2) n] + 1 = chi(r)(P-n) <= chi(*)(r)(Pn) <= 2 [log(2) n] + 1. In this paper it is proved that chi(*)(r)(P-n) > 1.619 log(2) n - 1. (c) 2006 Elsevier B.V. All rights reserved.