Lower bounds for on-line graph colorings

TL;DR

This paper introduces two strategies for Presenter in online graph coloring, establishing near-optimal lower bounds for coloring bipartite graphs and graphs excluding small cycles, highlighting fundamental limits of online algorithms.

Contribution

It presents two new strategies that establish tight lower bounds for online graph coloring in bipartite and cycle-free graphs, advancing understanding of algorithmic limitations.

Findings

For bipartite graphs, any online coloring algorithm requires at least 2 log2 n - 10 colors.

For graphs without triangles or pentagons, any online coloring algorithm needs at least Omega((n / log n)^{1/3}) colors.

Existing algorithms for these classes use significantly more colors, showing a gap between lower bounds and current methods.

Abstract

We propose two strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use $2\log_2 n - 10$ colors, where $n$ is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs graphs that contain neither $C_3$ nor $C_5$ as a subgraph and forces $\Omega(\frac{n}{\log n}^\frac{1}{3})$ colors. The best known on-line coloring algorithm for these graphs uses $O(n^{\frac{1}{2}})$ colors.