The orientation number of two complete graphs with linkages

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)}. In this paper, we study the orientation numbers of a family of graphs, denoted by G(p, q;m), that are obtained from the disjoint union of two complete graphs Kp and Kq by adding m edges linking them in an arbitrary manner. Define → d (m)=min{d (G): G ∈ G(p, q;m)}. We prove that d (2)= 4 and min{m: d (m)= 3} = 4. Let = min{m: d (m)= 2}. We evaluate the exact value of when p q p + 3 and show that 2p + 2 2p + 4 for q p + 4. © 2005 Elsevier B.V. All rights reserved.