Online Chromatic Number is PSPACE-Complete

TL;DR

This paper proves that determining the online chromatic number of a graph, where vertices arrive sequentially and must be colored immediately, is a PSPACE-complete problem, highlighting its computational complexity.

Contribution

It establishes the PSPACE-completeness of computing the online chromatic number, a significant complexity result in online graph coloring.

Findings

Computing the online chromatic number is PSPACE-complete.

The problem remains hard even with known graph structure.

Implications for online algorithms and complexity theory.

Abstract

In the online graph coloring problem, vertices from a graph G, known in advance, arrive in an online fashion and an algorithm must immediately assign a color to each incoming vertex v so that the revealed graph is properly colored. The exact location of v in the graph G is not known to the algorithm. The online chromatic number of G is the smallest number of colors such that some online algorithm is able to properly color G for any incoming order. We prove that computing the online chromatic number of a graph is PSPACE-complete.