Feedback edge set in bipartite digraph

RESUMO

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 \(\beta(G)\le \gamma(G)/2\). We also determine the exact Turán number of \(2k\)-free strong bipartite digraphs with partite sets \(X\) and \(Y\): if \(|X|,|Y|\ge k+1\), then the maximum number of edges is $$(|X|-(k-1))(|Y|-(k-1))+2k-2.$$ Finally, for the extremal case \(k=2\), we analyze the structure of \(4\)-free strong bipartite Turán digraphs and prove the sharper bound \(\beta(G)\le \gamma(G)/3\) for all such digraphs. This constant is attained by a natural balanced three-block construction.