Rainbow perfect domination in lattice graphs - Luis R. Fuentes; Italo J. Dejter; Carlos A. Araujo (Electronic Journal of Graph Theory and Applications (EJGTA), 2018)
Let 0 < n ∈ Z. In the unit distance graph of Zn ⊂ Rn, a perfect dominating set is understood as having induced components not necessarily trivial. A modification of that is proposed: a rainbow perfect dominating set, or RPDS, imitates a perfect-distance dominating set via a truncated metric; this has a distance involving at most once each coordinate direction taken as an edge color. Then, lattice-like RPDS s are built with their induced components C having: i vertex sets V(C) whose convex hulls are n-parallelotopes (resp., both (n − 1)- and 0-cubes) and ii each V(C) contained in a corresponding rainbow sphere centered at C with radius n (resp., radii 1 and n − 2).