PARTITION OF A BIPARTITE GRAPH INTO CYCLES

El-Zahar (1984) conjectured that if G is a graph on n1 + n2 + ... + n(k) vertices with n(i) greater-than-or-equal-to 3 for 1 less-than-or-equal-to i less-than-or-equal-to k and minimum degree delta(G) greater-than-or-equal-to [n1/2] + [n2/2] + ... + [n(k)/2], then G contains k vertex-disjoint cycles of lengths n1, n2, ..., n(k) respectively. He proved this conjecture for k = 2. In this note we consider a similar problem in bipartite graphs and prove: If G is a bipartite graph with partites V1 and V2 where \V1\ = \V2\ = n, n = n1 + n2 + ... + n(k) for n1 greater-than-or-equal-to n2 greater-than-or-equal-to ... greater-than-or-equal-to n(k) greater-than-or-equal-to 2, k greater-than-or-equal-to 2 and delta(G) greater-than-or-equal-to n1 + n2 + ... + n(k - 1) + n(k)/2, then G contains k vertex-disjoint cycles of lengths 2n1, 2n2, ..., 2n(k), respectively. We demonstrate by producing examples that the above result is best possible when k = 2.