Approach Vizing’s bound in two-party edge coloring

Determine how closely the palette size in two-party edge coloring can approach Vizing’s bound of [?] colors without substantially increasing the communication, including whether [?] or [?] edge coloring can be achieved with O(n) expected communication and whether sublinear communication is possible when the maximum degree grows with n.

Background

The paper constructs a public-coin Las Vegas protocol for proper (1+ε)Δ-edge coloring in the two-party edge-partition model, with expected communication O(ne{-γΔ}+1). This approaches the degree-dependent palette size guaranteed by Vizing’s theorem but does not reach Δ+1 or even Δ+O(1) colors.

The authors explicitly ask whether substantially fewer colors can be obtained without losing the communication advantages of their protocol, and whether the sublinear-communication phenomenon persists as the palette approaches the exact Vizing bound.

References

How far can the palette size be pushed toward Vizing's bound Δ+1 without substantially increasing the communication? Can one obtain a Δ+O(1) or Δ+1 edge coloring with O(n) expected communication? More ambitiously, is sublinear communication possible in this regime when Δ grows with n?

— 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 “Approaching Vizing's bound”

Can the sublinear communication achieved by our edge-coloring protocol when Δ=ω(1) also be obtained deterministically? Both our constructive LLL and the edge-coloring application use randomness in an essential way.

— 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 “Deterministic protocols”