Cyclability and pancyclability in bipartite graphs Original Research Article

Let G be a 2-connected bipartite balanced graph of order 2n and bipartition (X,Y). Let S be a subset of X of cardinality at least 3. We define S to be cyclable in G if there exists a cycle through all the vertices of S. Also, G is said S-pancyclable if for every integer l, 3⩽l⩽|S|, there exists a cycle in G that contains exactly l vertices of S. We prove that if the degree sum in G of every pair of nonadjacent vertices (x,y), x∈S, y∈Y is at least n+1, then S is cyclable in G. Under the same assumption where n+1 is replaced by n+3, we also prove that the graph G is S-pancyclable.