Rainbow perfect domination in lattice graphs

Luis R. Fuentes, Italo J. Dejter, Carlos A. Araujo

Abstract


Let $0<n\in\mathbb{Z}$. In the unit distance graph of $\mathbb{Z}^n\subset\mathbb{R}^n$, 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).


Keywords


perfect dominating set, lattice, tiling, rainbow sphere

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DOI: http://dx.doi.org/10.5614/ejgta.2018.6.1.7

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