On the bipartite case of El-Zahárs conjecture

A special case of a conjecture of El-Zahár states that a graph G with 2k vertices and minimum degree k, contains every bipartite 2-regular graph H on 2k vertices as a spanning subgraph. In this paper it will be proved that G contains the union of (k−1) cycles of lengths 4 and a path of order 4. Using this result it will also be proved that G contains the union of (k−2) cycles of lengths 4 and a cycle of length 8.