Feedback edge set in bipartite digraph

Let $\beta (G)$ denote the minimum size of a feedback edge set of a digraph $G$, and let $\gamma (G)$ denote the number of unordered pairs of nonadjacent vertices. Motivated by the Chudnovsky–Seymour–Sullivan conjecture for $3$-free digraphs, we study the corresponding feedback-edge problem for bipartite digraphs. In the bipartite setting, $\gamma (G)$ is taken to count only nonadjacent pairs with ends in distinct partite sets. We prove that every $4$-free bipartite digraph $G$ satisfies $\beta (G)\le \gamma (G)\mathrm{/}2$. We also determine the exact Turán number of $2k$-free strong bipartite digraphs with partite sets $X$ and $Y$: if $\mathrm{\mid }X\mathrm{\mid },\mathrm{\mid }Y\mathrm{\mid }\ge k+1$, then the maximum number of edges is $$(\mathrm{\mid }X\mathrm{\mid }-(k-1))(\mathrm{\mid }Y\mathrm{\mid }-(k-1))+2k-2.$$ Finally, for the extremal case $k=2$, we analyze the structure of $4$-free strong bipartite Turán digraphs and prove the sharper bound $\beta (G)\le \gamma (G)\mathrm{/}3$ for all such digraphs. This constant is attained by a natural balanced three-block construction.