On partitioning a bipartite graph into cycles and degenerated cycles

For a bipartite graph $G$, let ${\sigma }_{1,1}\left(G\right)$ be the minimum degree sum of two non-adjacent vertices in different partite sets of $G$. We prove the following result: If $G$ is a balanced bipartite graph of order $2n\ge k$, and if ${\sigma }_{1,1}\left(G\right)\ge n-\left((k-1)/2\right)+1$, then one of the following (i)–(iv) holds: (i) $G$ contains $k$ vertex-disjoint subgraphs $H_1,\dots ,H_k$ such that ${\bigcup }_{1\le i\le k}V\left(H_i\right)=V\left(G\right)$ and for each $i$, $1\le i\le k$, $H_i$ is a cycle or $K_1$ or $K_2$; (ii) $G\cong C_6$ and $k=2$; (iii) $G\cong C_8$ and $k=2,3$; (iv) $G\cong C_{10}$ and $k=4$. This result is a bipartite graph version of the result of Enomoto and Li (2004). We actually prove a stronger result which gives us control on the number of cycles in the $k$ vertex-disjoint subgraphs of (i).