Online Rainbow Coloring In Graphs

TL;DR

This paper introduces the first online algorithm for rainbow coloring in graphs, analyzing its performance across various graph types and establishing competitive ratios for different graph classes.

Contribution

It proposes the LRUC algorithm for online rainbow coloring and provides competitive analysis, including optimality and ratios for specific graph classes.

Findings

LRUC is optimal for line, tree, and star graphs.

LRUC has a competitive ratio of (2-2/n) for 1-cyclic graphs.

Competitive ratios for wheel and complete graphs are (n-1)/3 and n-1, respectively.

Abstract

Rainbow coloring is a special case of edge coloring, where there must be at least one path between every distinct pair of vertices that consists of different color edges. Here, we may use the same color for the adjacent edges of a graph representing two different paths from a single vertex. In online rainbow coloring, we have no priori knowledge about the vertices and edges of the graph, in fact the edges are available one by one. We have to color an edge as soon as it arrives and before the arrival of the next edge. We can not revoke the coloring decision once it is made. According to our knowledge, there is no study of online rainbow coloring for graphs. In this paper, we make a first attempt to propose an online algorithm named Least Recently Used Color(LRUC) for online rainbow coloring. We analyze the performance of LRUC through competitive analysis. We show that LRUC is optimal for line graph, tree and star graph. For 1-cyclic graph, LRUC is shown to be (2-2/n)-competitive, where n>3. We obtain the competitive ratios of (n-1)/3 and n-1 for wheel and complete graphs respectively, where n is the number of vertices.