The extremal edge-count conjecture for odd-order triangle-free 1-planar graphs

Let ${\cal P}_1$ denote the class of 1-planar graphs, and let $\operatorname{Forb}_{{\cal P}_1,n}(K_3)$ be the set of $n$-vertex 1-planar graphs containing no copy of $K_3$. Write

$$ex_{{\cal P}_1,n}(K_3)=\max\{e(G):G\in\operatorname{Forb}_{{\cal P}_1,n}(K_3)\}.$$

Extremal edge-count conjecture. For any odd $n\geq 5$,

$$ex_{{\cal P}_1,n}(K_3)=3n-9.$$

The theorem preceding this conjecture establishes the upper bound $ex_{{\cal P}_1,n}(K_3)\leq 3n-8$ for $n\geq 8$, with equality for even $n$. For odd $n\geq 9$, it is unknown whether the upper bound can be attained; the conjecture asserts that the exact extremal value is instead $3n-9$. Karpov constructed bipartite examples with $3n-9$ edges for every odd $n\geq 5$.