On-line rankings of graphs

AbstractA (vertex) k-ranking of a graph G=(V,E) is a proper vertex coloring ϕ:V→{1,…,k} such that each path with endvertices of the same color i contains an internal vertex of color ⩾i+1. In the on-line coloring algorithms, the vertices v1,…,vn arrive one by one in an unrestricted order, and only the edges inside the set {v1,…,vi} are known when the color of vi has to be chosen. We characterize those graphs for which a 3-ranking can be found on-line. We also prove that the greedy (First-Fit) on-line algorithm, assigning the smallest feasible color to the next vertex at each step, generates a (3log2n)-ranking for the path with n⩾2 vertices, independently of the order in which the vertices are received