On a conjecture concerning the orientation number of a graph

For a connected graph GG containing no bridges, let D(G)D(G) be the family of strong orientations of GG; and for any D∈D(G)D∈D(G), we denote by d(D)d(D) the diameter of DD. The orientation number d⃗(G) of GG is defined by d⃗(G)=min{d(D)|D∈D(G)}. Let G(p,q;m)G(p,q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs KpKp and KqKq by adding mm edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G(p,q;m)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 d⃗(m)=min{d⃗(G):G∈G(p,q;m)} and α=min{m:d⃗(m)=2}. In this paper we prove a conjecture on αα proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q≥p+4q≥p+4.