发表机构
Google DeepMind（谷歌DeepMind）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究聚焦小型锦标赛的$L_2$ Turán问题，确定了无$TT_4$、$R_4$等有向图的$L_2$范数平方最大值，补充了$Reg_5$的结果并证明了无$\boldsymbol{\text{的稳定性版本。
AI 中文摘要
我们研究各类小型有向图的$L_2$ Turán问题，重点关注自逆锦标赛及其稳定性版本。首先，我们确定了避开传递锦标赛$TT_4$和强连通锦标赛$R_4$的有向图的出度序列的$L_2$范数平方的精确最大值，解答了近期论文中的开放问题。我们证明，完全3部有向Turán图$T_3(m)$能精确最大化无$TT_4$有向图的$L_2$范数平方。对于无$R_4$有向图，最大值由$T_3(m)$取得，仅当$m \not\neq 1 \bmod 3$时例外，此时剥离一个终端汇点形成$T_3(m-1) \to v$会严格提升目标值。我们补充了正则锦标赛$Reg_5$的精确值并提出猜想。此外，我们证明了无$\boldsymbol{\text{有向3环}\boldsymbol{\text{（}\boldsymbol{\text{）}}}\boldsymbol{\text{的有向图的稳定性版本：任何渐近达到最大$L_2$密度的有向图序列，与极值有序双链$\boldsymbol{\text{（}\boldsymbol{\text{）}}}\boldsymbol{\text{的编辑距离为$O(\boldsymbol{\text{）}}}\boldsymbol{\text{）}}m^2$。
英文摘要
We investigate the $L_2$ Turán problems for various small directed graphs, specifically focusing on self-converse tournaments and stability versions. First, we determine the exact maximum $L_2$ norm squared of the out-degree sequence for digraphs avoiding the transitive tournament $TT_4$ and the strongly connected tournament $R_4$, answering open questions from recent paper. We prove that the complete directed 3-partite Turán graph $T_3(m)$ exactly maximizes the $L_2$ norm squared for $TT_4$-free digraphs. For $R_4$-free digraphs, the maximum is achieved by $T_3(m)$ except when $m \equiv 1 \pmod 3$, where peeling off a terminal sink vertex to form $T_3(m-1) \to v$ strictly increases the objective. We complement these results with exact values and a conjecture for the regular tournament $Reg_5$. Furthermore, we prove a stability version for $\vec{C}_3$-free digraphs: any sequence of digraphs asymptotically achieving the maximum $L_2$ density must have an edit distance of $O(δ^{1/2})m^2$ to the extremal ordered digon-chain $\vec{F}_{m,2}$.
