On Connected Turán Problems in Bipartite Graphs and Their Applications

We determine the exact extremal numbers and the structure of extremal graphs for connected bipartite graphs with fixed color-class sizes that avoid long paths. The proof relies on a structural characterization of 2-connected bipartite graphs without long cycles, combining Kopylov’s method with a strengthened Jackson lemma. Applications include concise proofs of classical results of Gyárfás–Rousseau–Schelp and Jackson on bipartite Turán numbers.