A note on the independence number in bipartite graphs

The independence number of a graph G, denoted by α(G), is the maximum cardinality of an independent set of vertices in G. The transversal number of G is the minimum cardinality of a set of vertices that covers all the edges of G. If G is a bipartite graph of order n, then it is easy to see that n 2 ≤ α(G) ≤ n − 1. If G has no edges, then α(G) = n = n(G). Volkmann [Australas. J. Combin. 41 (2008), 219– 222] presented a constructive characterization of bipartite graphs G of order n for which α(G) = n 2 . In this paper we characterize all bipartite graphs G of order n with α(G) = k, for each n 2 ≤ k ≤ n − 1. We also give a characterization on the Nordhaus-Gaddum type inequalities on the transversal number of trees.