Determine the degree-dependent communication complexity of edge coloring

Determine the tight communication complexity of (1+ε)Δ-edge coloring as a function of both the number of vertices n and the maximum degree Δ, and establish whether the upper-bound dependence ne^{-Ω(Δ)} is optimal.

Background

The main protocol achieves expected communication O(ne{-γΔ}+1) for (1+ε)Δ-edge coloring, while prior work gives an Ω(n) lower bound for constant-degree graphs. The resulting gap leaves the optimal dependence on Δ unresolved.

The open problem asks for matching upper and lower bounds across the full range of maximum degrees, rather than only for constant-degree instances.

References

A natural goal is therefore to determine the tight communication complexity as a function of both n and Δ. In particular, is the dependence ne{-Ω(Δ)} in our upper bound optimal?

— 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”