The degree-sum conjecture for feasible disjoint cycles in bipartite graphs

Let $G[X,Y]$ be a balanced bipartite graph of order $2n$, let $S$ be a subset of $X$ with $|S|\geq 2k+1$, and let $\sigma_{1,1}(S)$ denote the degree-sum parameter used in the paper. A collection of cycles is $S$-$k$-feasible when it consists of $k$ disjoint cycles whose vertices in $S$ collectively satisfy the required feasibility condition. Degree-sum conjecture. If $k$ is a positive integer and

$$\sigma_{1,1}(S)\geq n+2k-|S|+1,$$

then $G$ is $S$-$k$-feasible. The conjecture proposes a sharp sufficient degree-sum condition for covering specified vertices by disjoint cycles; the examples preceding it show that the bound and the hypothesis $|S|\geq 2k+1$ cannot be lowered in the ways considered there.