Lower and upper orientable strong radius and strong diameter of complete k-partite graphs ?
For two vertices u u and v v in a strong digraph D D , the strong distance sd(u,v) sd ( u , v ) between u u and v v is the minimum size (the number of arcs) of a strong sub-digraph of D D containing u u and v v . For a vertex v v of D D , the strong eccentricity se(v) se ( v ) is the strong distance between v v and a vertex farthest from v v . The strong radius srad(D) srad ( D ) (resp. strong diameter sdiam(D) sdiam ( D ) ) is the minimum (resp. maximum) strong eccentricity among the vertices of D D . The lower (resp. upper) orientable strong radius srad(G) srad ( G ) (resp. SRAD(G) SRAD ( G ) ) of a graph G is the minimum (resp. maximum) strong radius over all strong orientations of G . The lower (resp. upper) orientable strong diameter sdiam(G) sdiam ( G ) (resp. SDIAM(G) SDIAM ( G ) ) of a graph G is the minimum (resp. maximum) strong diameter over all strong orientations of G . In this paper, we determine the lower orientable strong radius and diameter of complete k k -partite graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of complete k k -partite graphs. We also find an error about the lower orientable strong diameter of complete bipartite graph Km,n K m , n given in [Y.-L. Lai, F.-H. Chiang, C.-H. Lin, T.-C. Yu, Strong distance of complete bipartite graphs, The 19th Workshop on Combinatorial Mathematics and Computation Theory, 2002, pp. 1216], and give a rigorous proof of a revised conclusion about sdiam(Km,n) sdiam ( K m , n ) .
lower and upper orientable strong radius and strong diameter strong distance