Degree Sum Conditions for Cyclability in Bipartite Graphs

We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d (G) (v) be the degree of a vertex v in a graph G. For G[X, Y] and we define . Amar et al. (Opusc. Math. 29:345-364, 2009) obtained sigma (1,1)(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| a parts per thousand yen |Y| and such that sigma (1,1)(S) a parts per thousand yen |X| + 1, then either there exists a cycle containing S or and there exists a cycle containing Y. This degree sum condition is sharp.