TOTAL MUTUAL-VISIBILITY IN HAMMING GRAPHS

If G is a graph and X C V (G), then X is a total mutual-visibility set if every pair of vertices x and y of G admits the shortest x, y-path P with V (P) n X C {x, y}. The cardinality of the largest total mutual-visibility set of G is the total mutual-visibility number mu t(G) of G. In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values mu t(Kn(1) square Kn(2) square Kn(3)) are determined. It is proved that mu t(Kn(1) square center dot center dot center dot square Kn(r) ) = O(Nr-2), where N = n1 + center dot center dot center dot +nr, and that mu t(K-s square ,r) = Theta(s(r-2)) for every r> 3, where K square ,r s denotes the Cartesian product of r copies of Ks. The main theorems are also reformulated as Tur & aacute;n-type results on hypergraphs.