Determine whether vertex-coloring communication decreases with maximum degree

Determine whether the randomized communication complexity of Δ+1 vertex coloring decreases as the maximum degree Δ increases, despite the known Θ(n) complexity in general and the fact that the existing linear lower bound is witnessed by bounded-degree graphs.

Background

The paper contrasts its degree-sensitive edge-coloring upper bound with the known Θ(n) randomized communication complexity of (Δ+1)-vertex coloring. The existing lower bound for vertex coloring is established using bounded-degree graphs, so it does not resolve whether higher degree permits lower communication.

The authors leave open whether vertex coloring exhibits a degree-dependent communication reduction analogous to the one obtained for edge coloring.

References

It remains open whether the communication complexity decreases with Δ for vertex coloring as well.

— Communication-Efficient $(1+\varepsilon)Δ$-Edge Coloring and Lovász Local Lemma  (2609.39359 - Chang et al., 30 Sep 2026) in Section 5, Conclusion and open problems, paragraph “Communication as a function of the degree”