An Optimal Algorithm for Conflict-Free Coloring for Tree of Rings

TL;DR

This paper introduces an optimal conflict-free coloring algorithm for trees of rings, reducing the number of colors used from previous algorithms while ensuring the unique-min property, which guarantees the minimum color is unique.

Contribution

It presents a new algorithm that improves color efficiency for conflict-free coloring in trees of rings, achieving optimal color usage with the unique-min property.

Findings

Uses O(log|T|*log|R|) colors, improving over previous algorithms

Ensures the coloring has the unique-min property

Achieves optimal conflict-free coloring for trees of rings

Abstract

An optimal algorithm is presented about Conflict-Free Coloring for connected subgraphs of tree of rings. Suppose the number of the rings in the tree is |T| and the maximum length of rings is |R|. A presented algorithm in [1] for a Tree of rings used O(log|T|.log|R|) colors but this algorithm uses O(log|T|+log|R|) colors. The coloring earned by this algorithm has the unique-min property, that is, the unique color is also minimum.