Characterization of tournaments with multipartite extremizers

Determine precisely which tournaments F have the property that ex_2^+(n,F) is attained uniquely by a balanced complete bidirected multipartite digraph.

Background

The paper studies the degree-square Turán number ex_2+(n,F), defined as the maximum sum of squared out-degrees over n-vertex F-free digraphs. It proves exact formulas and uniqueness of the balanced complete bidirected multipartite extremizer for all transitive tournaments TT_r and for the regular tournament RT on five vertices.

The concluding remarks ask whether this extremal phenomenon extends to other forbidden tournaments and seek a classification of all tournaments for which the balanced complete bidirected multipartite construction is the unique extremal digraph. This question is left unresolved beyond the tournament families treated in the paper.

References

It remains natural to ask whether the same phenomenon occurs for other forbidden tournaments. That is, for which tournaments $F$ is $ex_2+(n,F)$ attained uniquely by a balanced complete bidirected multipartite digraph?

— Degree-square Turán problem for two self-converse tournaments  (2610.01047 - Han et al., 1 Oct 2026) in Section 6, Concluding remarks