Maximal edge-connectivity in the odd case of Theorem 7

Strengthen Theorem 7 by proving that the resulting equitable l-colorable realization is maximally edge-connected when the relevant minimum-degree parity case is odd, while preserving the realization constraints and the subgraph structure specified in Theorem 7.

Background

Theorem 7 constructs an equitable l-colorable realization G from an equitable l-colorable graph G₀ with a spanning subgraph Z₀, guaranteeing edge-connectivity δ(G)−1 when δ(G) is odd and at least 3, and maximal edge-connectivity in the other stated cases.

The authors identify a specific unresolved strengthening: obtaining maximal edge-connectivity also in the odd case. They explain that the obstacle arises because a proof step permits one of two possible edge exchanges, but equitable colorability cannot currently be guaranteed for both choices.