Cyclability and pancyclability in bipartite graphs

Abstract Let G be a 2-connected bipartite balanced graph of order 2 n 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.