Vertex-Disjoint Cycles Containing Specified Vertices in a Bipartite Graph

Abstract

The theory of vertex-disjoint cycles of a graph is the generalization of the well-known Hamiltonian cycle theory. In this paper, we prove the following result. Let $G = (V_{1}, V_{2}; E$) be a bipartite graph with $|V_{1}|= |V_{2}|= n$ such that $n\ge 2k + 1$, where k $\ge$ 1 is an integer. If $\sigma _{1,1}(G) \ge n + k$, then for any k distinct vertices $v_{1}, v_{2}, \ldots , v_{k}$ of G, G contains $k - 1$ quadrilaterals $C_{1}, C_{2}, \ldots , C_{k-1}$ and a path $P_{k}$ of order 2t, where $t = n - 2(k - 1)$, such that all of them are vertex-disjoint and $v_{i} \in V(C_{i})$ for each $i \in \{1, 2, \ldots , k - 1\}, v_{k} \in V(P_{k})$. Using this result we also prove that G contains k vertex-disjoint cycles $C_{1}, C_{2}, \ldots , C_{k}$ such that $v_{i} \in V(C_{i})$ for each $i \in \{1, 2, \ldots , k\}$ and there are $k - 1$ quadrilaterals in $\{C_{1}, C_{2}, \ldots , C_{k}\}$. Moreover, the degree condition is sharp.