On-line ranking number for cycles and paths

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) < 3 log2 n for n ≥ 2. Here we show that χ∗r (Pn) ≤ 2blog2 nc+1. The same upper bound is obtained for χ∗ r (Cn), n ≥ 3.