Characterization of graphs with discrepancy at most one

Determine precisely which graphs G and positive integers k satisfy D_k(G)≤1, where D_k(G) is the minimum, over all edge k-colourings of G, of the maximum difference between the numbers of edges of any two colours incident with any vertex.

Background

The paper defines the k-discrepancy D_k(G) as a local equitable edge-colouring parameter and proves the universal upper bound D_k(G)≤2 for every graph and every positive integer k.

The authors then leave open the sharper classification problem of identifying exactly when the value can be reduced from 2 to 1.