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 Kq 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. c © 2008 Elsevier B.V. All rights reserved.