Bi-cycle extendable through a given set in balanced bipartite graphs

Abstract

Let G = ( X , Y ; E ) be a balanced bipartite graph of order 2 n . The path-cover number p c ( H ) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H . S ⊆ V ( G ) is called a balanced set of G if | S ∩ X | = | S ∩ Y | . In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S , there is a bi-cycle of every length from | S | + 2 p c ( 〈 S 〉 ) up to 2 n through S .