Lower and upper orientable strong radius and strong diameter of complete k-partite graphs

Abstract

For two vertices u and v in a strong digraph D , the strong distance sd ( 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 se ( v ) is the strong distance between v and a vertex farthest from v . The strong radius srad ( D ) (resp. strong diameter sdiam ( D ) ) is the minimum (resp. maximum) strong eccentricity among the vertices of D . The lower (resp. upper) orientable strong radius srad ( G ) (resp. 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 ) (resp. 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 -partite graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of complete k -partite graphs. We also find an error about the lower orientable strong diameter of complete bipartite graph 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. 12–16], and give a rigorous proof of a revised conclusion about sdiam ( K m , n ) .