The following problem was solved by Woodall in 1972: for any pair of nonnegative integers n and k < n/2 - 1 find the minimum integer g(n, k) such that every graph with n vertices and at least g(n, k) edges contains a cycle of length n - k. Woodall proved even more: the size g(n, k), in fact, guarantees the existence of cycles C, for all 3 <= p <= n - k. <br> <br> In the paper an analogous problem for bipartite graphs is considered. It is proved that every bipartite graph with color classes of cardinalities m and n, m <= n, and size greater than n(m - k - 1) + k + 1 contains a cycle of length 2m - 2k, where m >= 1/2k(2) + 3/2k + 4, k is an element of N. The bound on the number of edges is best possible. Moreover, this size condition guarantees the existence of cycles of all even lengths up to 2m - 2k. We also characterize all extremal graphs for this problem. Finally, we conjecture that the condition on the order may be relaxed to m >= 2k + 2.
Discrete Mathematics & Theoretical Computer Science