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 $$G = (V_{1}, V_{2}; E$$G=(V1,V2źE) be a bipartite graph with $$|V_{1}|= |V_{2}|= n$$|V1|=|V2|=n such that $$n\ge 2k + 1$$nź2k+1, where k$$\ge $$ź 1 is an integer. If $$\sigma _{1,1}(G) \ge n + k$$ź1,1(G)źn+k, then for any k distinct vertices $$v_{1}, v_{2}, \ldots , v_{k}$$v1,v2,ź,vk of G, G contains $$k - 1$$k-1 quadrilaterals $$C_{1}, C_{2}, \ldots , C_{k-1}$$C1,C2,ź,Ck-1 and a path $$P_{k}$$Pk of order 2t, where $$t = n - 2(k - 1)$$t=n-2(k-1), such that all of them are vertex-disjoint and $$v_{i} \in V(C_{i})$$viźV(Ci) for each $$i \in \{1, 2, \ldots , k - 1\}, v_{k} \in V(P_{k})$$iź{1,2,ź,k-1},vkźV(Pk). Using this result we also prove that G contains k vertex-disjoint cycles $$C_{1}, C_{2}, \ldots , C_{k}$$C1,C2,ź,Ck such that $$v_{i} \in V(C_{i})$$viźV(Ci) for each $$i \in \{1, 2, \ldots , k\}$$iź{1,2,ź,k} and there are $$k - 1$$k-1 quadrilaterals in $$\{C_{1}, C_{2}, \ldots , C_{k}\}$${C1,C2,ź,Ck}. Moreover, the degree condition is sharp.