Extended Online Graph Edge Coloring

Abstract

Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The condition which we follow in graph coloring is that the incident vertices/edges have not the same color. There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and others are online algorithm. Where offline means the graph is known in advance and the online means that the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is added to the graph and the main issue is that we want to minimize the number of colors. We cannot change the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is Δ, then it is possible to color its edges, in polynomial time, using at most Δ+ 1 color. The algorithm provided by Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors online in multiple rounds.