Geodesic
3-Tuple Total Domination Number of Rook’s Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 15-37

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A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S. The minimum size of a kTDS is called the k-tuple total dominating number and it is denoted by γ×k,t(G). We give a constructive proof of a general formula for γ×3,t(Kn□Km).
Keywords: k -tuple total domination, Cartesian product of graphs, rook’s graph, Vizing’s conjecture 
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     title = {3-Tuple {Total} {Domination} {Number} of {Rook{\textquoteright}s} {Graphs}},
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Pahlavsay, Behnaz; Palezzato, Elisa; Torielli, Michele. 3-Tuple Total Domination Number of Rook’s Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 15-37. http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a1/