Degree-square Turán problem for two self-converse tournaments

Abstract

For a fixed digraph $F$, let ${\mathrm{ex}}_2^{+}(n,F)$ be the maximum of ${\sum }_{v\in V(D)}{d}_D^{+}(v{)}^2$ over all $n$-vertex $F$-free digraphs. Ai et al. [arXiv:2606.03520, 2026] asked for which self-converse tournament $F$ one can determine ${\mathrm{ex}}_2^{+}(n,F)$. Let $T{T}_r$ denote a transitive tournament on $r$ vertices and ${\text{RT}}_5$ denote a regular tournament on 5 vertices. Note that both $T{T}_r$ and ${\text{RT}}_5$ are self-converse. Iľkovič [arXiv:2609.05042, 2026] determined ${\mathrm{ex}}_2^{+}(n,F)$ for $F=T{T}_4$. In this paper, we determine ${\mathrm{ex}}_2^{+}(n,F)$ for $F$ being $T{T}_r$ or ${\text{RT}}_5$, and the latter confirms a conjecture due to Iľkovič.

arXiv comment20 pages, no figures