Extremal graph for bowtie orientations of types (a) and (b)

Prove that, for every orientation B of the bowtie graph F_{2,3} having either all edges incident with the center oriented uniformly or the two triangles oriented oppositely as in Cases (a) or (b) of Proposition \ref{not_semi_anti_directed_prop}, a complete tripartite graph with part sizes 1,(n-1)/2,(n-1)/2 attains D(n,B).

Background

The bowtie B is the graph F_{2,3}, consisting of two triangles sharing a center vertex. Proposition \ref{not_semi_anti_directed_prop} identifies orientations in Cases (a) and (b) for which D(n,B)>2{ex(n,B)}. The conjecture proposes that the maximizer is nevertheless a complete tripartite graph with one singleton part and two balanced parts, specifying the extremal host graph for these unresolved orientations.

References

We begin with some conjectures about the unresolved orientations of $B = F_{2,3}$. First, we conjecture the value of $D(n, B)$ for the orientations mentioned in Proposition \ref{not_semi_anti_directed_prop}.

— Orientations of graphs omitting non-edge-critical directed graphs  (2502.21287 - Sheats, 28 Feb 2025) in Section 5, subsection “Questions and open problems,” Conjecture \ref{all_bowties_conj}

First, we conjecture the value of D(n, B) for the orientations mentioned in Proposition \ref{not_semi_anti_directed_prop}. Let G be a complete tripartite graph with part sizes 1, {\frac{n-1}{2}, and {\frac{n-1}{2}. Let B be an orientation of B as in Case (a) or (b) of Proposition \ref{not_semi_anti_directed_prop}. Then $$D(G, B) = D(n, B).$$

— Orientations of graphs omitting non-edge-critical directed graphs  (2502.21287 - Sheats, 28 Feb 2025) in Section 5, subsection “Questions and open problems,” Conjecture labeled all_bowties_conj