Extremal graph for non-semi-anti-directed bowtie orientations

Determine whether, for every orientation B of the bowtie graph F_{2,3} covered by Case (a) or Case (b) of Proposition 5.1, a complete tripartite graph with part sizes 1, (n-1)/2, and (n-1)/2 maximizes the number D(G,B) of B-free orientations among all n-vertex graphs.

Background

The paper studies D(G,H), the number of orientations of the edges of an undirected graph G that avoid a specified directed graph H, and D(n,H), the maximum of this quantity over all n-vertex graphs. For the bowtie graph B=F_{2,3}, Proposition \ref{not_semi_anti_directed_prop} identifies orientations in two cases for which D(n,B) is strictly larger than the standard extremal lower bound 2{ex(n,B)}.

The conjecture proposes the extremal host graph for these orientations: a complete tripartite graph with one singleton part and two approximately equal parts. Establishing the conjecture would identify the graph attaining D(n,B) for these unresolved bowtie 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}. 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 \ref{all_bowties_conj}