Rainbow perfect domination in lattice graphs

Luis R. Fuentes (University of Puerto Rico Rio Piedras, PR 00936-8377 Puerto Rico)
Italo J. Dejter (University of Puerto Rico Rio Piedras, PR 00936-8377 Puerto Rico)
Carlos A. Araujo (Universidad del Atlantico Km 7 Via Puerto Barranquilla Colombia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 6, No 1 (2018): Electronic Journal of Graph Theory and Applications

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).