Vertex-disjoint cycles containing specified edges in a bipartite graph (Q2712508)

The main result is a minimum degree condition and a sum of degree condition on a bipartite graph that implies the existence of a \(2\)-factor with each cycle of the \(2\)-factor containing a specified edge. In particular it is shown that if \(k \geq 2\), \(n \geq 2k\), and either NEWLINE\[NEWLINE \sigma_{1,1} (G) \geq \max\left\{n + k, \left\lceil \frac{2n-1}{3} \right\rceil + 2k\right\} \quad \text{or} \quad \delta (G) \geq \max\left\{\frac{n + k}{2}, \left\lceil \frac{2n+4k}{5} \right\rceil\right\}, NEWLINE\]NEWLINE then for any independent edges \(e_1, e_2, \dots, e_k\), the bipartite graph \(G\) can be partitioned into cycles \(H_1, H_2, \dots, H_k\) such that \(e_i \in E(H_i)\). In this result \(\sigma_{1,1}\) is the minimum sum of degrees of nonadjacent vertices in different parts of the bipartite graph. This result parallels a \(2\)-factor containing specified edges result by \textit{Y. Egawa} et al. [Graphs Comb. 16, No. 1, 81-92 (2000; Zbl 0951.05061)] for graphs in general.