On-line rankings of graphs

Abstract

A (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 v 1,…, v n arrive one by one in an unrestricted order, and only the edges inside the set { v 1,…, v i } are known when the color of v i 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 (3 log 2 n) -ranking for the path with n⩾2 vertices, independently of the order in which the vertices are received.