$L_2$ Turán Problems for Small Tournaments and Stability

Abstract

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 $T{T}_4$ 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 $T{T}_4$-free digraphs. For $R_4$-free digraphs, the maximum is achieved by $T_3(m)$ except when $m\equiv 1(\mathrm{mod}3)$, 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 $Re{g}_5$. 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 $O({\delta }^{1/2})m^2$ to the extremal ordered digon-chain ${\vec{F}}_{m,2}$.