# Tight Bounds for Online Coloring of Basic Graph Classes

## TL;DR

This paper establishes tight lower bounds on online graph coloring algorithms for various fundamental graph classes, demonstrating that the optimal competitive ratio is logarithmic in the number of vertices, and showing the effectiveness of the First Fit algorithm.

## Contribution

It provides the first tight bounds for online coloring of multiple graph classes, including randomized algorithms, and shows First Fit's near-optimality in these settings.

## Key findings

- Optimal competitive ratio is Θ(log n) for various graph classes.
- First Fit algorithm achieves near-optimal performance in these classes.
- Bounds hold even with lookahead or reordering buffers.

## Abstract

We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is $\Theta(\log n)$, where $n$ is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size $O(n/\log n)$ or access to a reordering buffer of size $n^{1-\epsilon}$, for any $0<\epsilon\leq 1$. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural $\textit{First Fit}$ coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized online algorithms.