Hardness of Approximating Continuous Chromatic Number

Determine whether the n^(1−ε) inapproximability of the classical chromatic number persists for the one-half-chromatic number problem on continuous graphs.

Background

The classical chromatic number is hard to approximate within n1−ε for every ε>0. The paper asks whether the same hardness persists when graph coloring is reformulated as coloring a radius-one-half ball cover of a continuous graph.

References

The question for the continuous version of the chromatic number problem remains as above: Whether or not the same hardness of approximation remains for the $\frac{1}{2}$-chromatic number problem on continuous graphs?

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Chromatic number,” paragraph “Approximability and bounds”