The $Reg_5$-free $L_2$ Turán conjecture

Prove that, for every positive integer $m$, the complete directed 4-partite Turán graph $T_4(m)$ maximizes the squared $L_2$ norm of the out-degree sequence among all $Reg_5$-free digraphs on $m$ vertices, equivalently that $\operatorname{ex}^+_{L_2}(m,Reg_5)=\|d^+_{T_4(m)}\|_2^2$.

Background

The paper studies extremal squared L2L_2 norms of out-degree sequences in digraphs avoiding specified tournaments. It proves exact extremal results for the transitive tournament TT4TT_4 and the strongly connected tournament R4R_4.

For the regular tournament Reg5Reg_5, the authors report exact computations for small values of mm and formulate the conjecture that the balanced complete directed 4-partite Turán graph T4(m)T_4(m) is always extremal. The conjecture includes explicit formulas according to the residue class of mm modulo 4, but no general proof is provided.

References

For all $m \ge 1$, the maximum $L_2$ norm squared of the out-degree sequence for a $Reg_5$-free digraph is exactly achieved by the Turán graph $T_4(m)$. That is, $\text{ex}+_{L_2}(m, Reg_5) = |d+_{T_4(m)}|_22$.

— $L_2$ Turán Problems for Small Tournaments and Stability  (2609.05042 - Iľkovič, 4 Sep 2026) in Conjecture in Section 3.1, “Exact Results for Small Tournaments”