Characterization of minimum-likelihood graphs for orders at least 15

Determine the graphs attaining the minimum likelihood among all graphs of order n for every n≥15, including whether minimizers are always blow-ups and whether their base graphs can be selected from a fixed finite family or must have orders that grow with n.

Background

The paper disproves the previously proposed conjecture that balanced complete bipartite graphs minimize likelihood for all n≥6: the first counterexample occurs at n=15, where the blow-up C₅[\overline{K₃}] has smaller likelihood than K₇,₈. The paper also proves that counterexamples occur for every sufficiently large order and provides many further counterexamples among balanced or nearly balanced blow-ups of C₅.

Despite these results, the global minimizer is not characterized for orders n≥15. In particular, the paper does not determine whether the n=15 counterexample C₅[\overline{K₃}] is itself globally minimizing, whether all minimizers belong to the class of graph blow-ups, or whether a bounded collection of base graphs suffices to describe the minimizers asymptotically.