Sharpness of the universal linear lower bound

Characterize the graphs F without an isolated edge for which the rainbow saturation number satisfies rsat(n,F)=\frac{\eta(F)}{2}n+o(n).

Background

Theorem 1.2 (Theorem TLower) establishes the universal lower bound rsat(n,F)\geq \frac{\eta(F)}{2}n-O(1) for every graph F without an isolated edge, where \eta(F) is the directed-weight parameter defined in Section 2. The concluding section asks for a characterization of exactly those graphs for which this lower bound is asymptotically attained, thereby identifying the sharpness class of the bound.