A CHARACTERIZATION OF BIPARTITE GRAPHS WITH GIVEN TRANSVERSAL NUMBER

Abstract. A transversal in G is a set of vertices that covers all the edges of G. The transversal number of G, denoted by τ(G), is the minimum cardinality of a transversal in G. If G is a bipartite graph of order n, then it is easy to see that 1 ≤ τ(G) ≤ n/2. If G has no edges, then τ(G) = 0. Volkmann in 2008, presented a constructive characterization of bipartite graphs G of order n for which τ(G) = ⌊ n/2 ⌋ . In this paper we characterize all bipartite graphs G of order n with τ(G) = k, for each 1 ≤ k ≤ ⌊ n/2 ⌋ . We also give a characterization on the Nordhaus-Gaddum type inequalities on the transversal number of trees.