Characterization of colored graphs with minimal face counts

Develop a complete characterization of D-colored graphs that minimize either the total number of faces F or the color-specific face counts F_c, beyond currently known examples (e.g., maximally single-trace graphs and certain partial-transpose moments).

Background

Several combinatorial quantities controlling state-distinguishing power—such as the total number of faces F and the per-color face counts F_c—play a central role in the paper’s analysis. While certain families (e.g., maximally single-trace graphs) are known to minimize F in specific senses, the authors note that a general classification is lacking.

A full structural description of these minimal-face graphs would clarify which trace-invariants can or cannot distinguish various reference states and would systematize many examples currently handled case-by-case.