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Total mutual-visibility in Hamming graphs

Bujtás Csilla, Klavžar Sandi, Tian Jing

Opuscula Mathematica

2025

Vol. 45, no. 1

63--78

If G is a graph and X ⊆ 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) ∩ X ⊆ {x, y}. The cardinality of the largest total mutual-visibility set of G is the total mutual-visibility number μt(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 μt(Kn1 □Kn2 □Kn3 ) are determined. It is proved that μt(Kn1 □ · · · □Knr ) = O(Nr−2), where N = n1 + · · · + nr, and that μt(Ks□,r) = Θ(sr−2) for every r ≥ 3, where Ks□,r denotes the Cartesian product of r copies of Ks. The main theorems are also reformulated as Turán-type results on hypergraphs.