On optimal orientation of cycle vertex multiplications

For a bridgeless connected graph G, let D(G) be the family of its strong orientations; and for any D∈D(G), we denote by d(D) its diameter. The orientation number d→(G) of G is defined by d→(G)=min{d(D)|D∈D(G)}. For a connected graph G of order n and for any sequence of n positive integers (si), let G(s1,s2,…,sn) denote the graph with vertex set V* and edge set E* such that V*=⋃i=1nVi, where Vi's are pairwise disjoint sets with |Vi|=si, i=1,2,…,n, and for any two distinct vertices x, y in V*, xy∈E* if and only if x∈Vi and y∈Vj for some i,j∈{1,2,…,n} with i≠j such that vivj∈E(G). We call the graph G(s1,s2,…,sn) a G vertex multiplication. In this paper, we determine the orientation numbers of various cycle vertex multiplications.