On a conjecture concerning the orientation number of a graph

For a connected graph G containing no bridges, let D ( G ) be the family of strong orientations of G ; and for any D ∈ D ( G ) , we denote by d ( D ) the diameter of D . The orientation number d ⃗ ( G ) of G is defined by d ⃗ ( G ) = min { d ( D ) | D ∈ D ( G ) } . Let G ( p , q ; m ) denote the family of simple graphs obtained from the disjoint union of two complete graphs K p and K q by adding m edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G ( p , q ; m ) was introduced by Koh and Ng [K.M. Koh, K.L. Ng, The orientation number of two complete graphs with linkages, Discrete Math. 295 (2005) 91–106]. Define d ⃗ ( m ) = min { d ⃗ ( G ) : G ∈ G ( p , q ; m ) } and α = min { m : d ⃗ ( m ) = 2 } . In this paper we prove a conjecture on α proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q ≥ p + 4 .