Optimal orientations of vertex-multiplications of trees with diameter 4
Koh and Tay proved a fundamental classi cation of G vertex-multiplications into three classes C0; C1 and C2. They also showed that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class C2. Of interest, G vertex-multiplications are extensions of complete n-partite graphs and Gutin characterised complete bipartite graphs with orientation number 3 (or 4 resp.) via an ingenious use of Sperner's theorem. In this paper, we investigate vertex-multiplications of trees with diameter 4 in C0 (or C1) and exhibitits intricate connections with problems in Sperner Theory, there by extending Gutin's approach. Let s denote the vertex-multiplication of the central vertex. We almost completely characterise the case of even s and give a complete characterisation for the case of odd s ≥ 3.