On the Fon-Der-Flaass interpretation of extremal examples for Turán’s (3, 4)-problem

AbstractFon-Der-Flaass (1988) presented a general construction that converts an arbitrary $\vec C_4 $-free oriented graph Γ into a Turán (3, 4)-graph. He observed that all Turán-Brown-Kostochka examples result from his construction, and proved the lower bound $\tfrac{4} {9} $ (1 − o(1)) on the edge density of any Turán (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound $\tfrac{3} {7} $ (1 − o(1)) on the edge density of any Turán (3, 4)-graph resulting from the Fon-Der-Flaass construction under any of the following assumptions on the undirected graph G underlying the oriented graph Γ: (i) G is complete multipartite; (ii) the edge density of G is not less than $$\tfrac{2} {3} - \varepsilon $$ for some absolute constant ε > 0. We are also able to improve Fon-Der-Flaass’s bound to $\tfrac{7} {{16}} $ (1 − o(1)) without any extra assumptions on Γ.