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 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 for n ≥ 2. Here we show...