The optimal strong radius and optimal strong diameter of the Cartesian product graphs

Let D be a strong digraph. The strong distance between two vertices u and v in D , denoted by s d D ( u , v ) , is the minimum size (the number of arcs) of a strong subdigraph 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 minimum strong eccentricity among all vertices of D is the strong radius, denoted by s r a d ( D ) , and the maximum strong eccentricity is the strong diameter, denoted by s d i a m ( D ) . The optimal strong radius (resp. strong diameter) s r a d ( G ) (resp. s d i a m ( G ) ) of a graph G is the minimum strong radius (resp. strong diameter) over all strong orientations of G . Juan et al. (2008) [Justie Su-Tzu Juan, Chun-Ming Huang, I-Fan Sun, The strong distance problem on the Cartesian product of graphs, Inform. Process. Lett. 107 (2008) 45–51] provided an upper and a lower bound for the optimal strong radius (resp. strong diameter) of the Cartesian products of any two connected graphs. In this work, we determine the exact value of the optimal strong radius of the Cartesian products of two connected graphs and a new upper bound for the optimal strong diameter. Furthermore, these results are also generalized to the Cartesian products of any n ( n > 2 ) connected graphs.