The study of rainbow coloring of graphs and graph coloring in streaming

Abstract

Graph coloring is a well known problem with wide-ranging applications. The vertex and edge coloring problems have been studied in various models of computation. Rainbow coloring is a type of edge coloring that also acts as a connectivity measure for graphs. A graph is said to be rainbow colored or rainbow connected if there exists an edge coloring such that every pair of vertices, if connected, is connected by a path having distinct colors for all edges contained in it. The veri cation of rainbow coloring is an NP-Complete problem whereas the problems of verifying vertex and edge coloring admit easy solutions in the RAM model. Veri cation of graph coloring in the streaming model of computation is a problem that has not been studied before. We focus on the vertex coloring problem in the streaming model and give algorithms that verify if a given vertex coloring is valid with a high probability. We also give lower bounds for verifying vertex coloring in a few streaming models.