Efficient total domination and related invariants in torus graphs

For a graph G, a set of vertices S is an efficient total k-dominating set (ETkD set) if every vertex of G is adjacent to exactly k vertices in S. A torus graph is the Cartesian product Cm□Cn of two cycles.In this work, we characterize all torus graphs with an ETkD set for k = 1, 2, 3. In particular, we prove following conjecture of Kuziak, Peterin, and Yero (2014): Cm□Cn has an ET1D set if and only if m ≡ n ≡ 0 (mod 4). This allows us to determine the total domatic number and the injective chromatic number of all torus graphs. Additionally, we characterize the signed total Roman domatic number of 4-regular graphs in terms of ET1D and ET2D sets, and consequently obtain exact values for all torus graphs.