Optimal strong [formula omitted]-orientation of complete [formula omitted]-partite graphs

Abstract

For two vertices u and v in a strong oriented graph D , the strong distance s d ( u , v ) between u and v is the minimum size (the number of arcs) of a strong sub-digraph of D containing u and v . For a vertex v of D , the strong eccentricity s e ( v ) is the strong distance between v and a vertex farthest from v . The strong diameter s d i a m ( D ) is the maximum strong eccentricity among the vertices of D . The lower orientable strong diameter s d i a m ( G ) of a graph G is the minimum strong diameter over all strong orientations of G . An orientation D of a graph G is said to be an optimal strong ( κ , d ) -orientation of G if κ ( D ) = ⌊ κ ( G ) / 2 ⌋ and s d i a m ( D ) = s d i a m ( G ) , where κ ( D ) (resp. κ ( G ) ) is the strong connectivity of D (resp. connectivity of G ). In this paper, we will show that each complete k -partite graph has an optimal strong ( κ , d ) -orientation.