Exact bipartite Turán numbers of large even cycles

Abstract The bipartite Turán number, denoted by , is the maximum number of edges in an ‐free bipartite graph with two parts of sizes and , respectively. In this article, we prove for any positive integers with , which confirms (in a strong form) the unsolved case of a conjecture of Győri. Let be a bipartite graph with and . Suppose that for some . We prove that contains cycles of all even lengths from 4 to if . This result sharpens a theorem on long cycles in bipartite graphs due to Jackson. As a main tool, for a longest cycle in a bipartite graph, we prove an upper bound of the number of edges which are incident with at most one vertex in , which is a bipartite analogue of a classical theorem of Bondy. Our proof technique is structural.