Rainbow perfect domination in lattice graphs

Abstract

Let 0 < n ∈ Z. In the unit distance graph of Z n ⊂ 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).

Truncated-distance codes and ternary compounds

“…We will say for G = Γ n in Sections 3-4 that: an edge-disjoint union G of triangles has a non-isolated PDS S if every vertex of G not in S is adjacent to just either one vertex of G (as in PDS's) or the two end-vertices of an edge of G. Section 4 shows that Γ 2 has an infinite number of isolated 2-PTDC and conjectures that Γ n has isolated n-PTDC's, ∀n > 2. Remark 2 We restate some results of [12] in Theorem 2, (resp. Theorems 3-4) below.…”

“…Extending the meaning of the adjective lattice in Subsection 1.2 below, as in [12] we have in Theorems 3-4 that a lattice κ-PTDC S exists in Λ n whose induced subgraph [S] has its components H with vertex sets V (H):…”

“…Section 2 contains a restatement of results of [12], translating its graphcoloring context into our present one. In fact, in Theorem 3 below, a lattice 1-PTDC[H 0 , H 1 ; m 0 , m 1 ] is obtained; in addition, a family of lattice (t 0 , t 1 )-PTDC[H 0 , H 1 ; m 0 , m 1 ]'s in the graphs Λ n that extend the lattice 1-PTDC of Theorem 3 is obtained in Theorem 4.…”

“…A natural follow-up here would be to consider lattice tilings of Z n with two different generalized Lee spheres. Focus on such follow-up for n > 3 is only possible by modifying the notion of the Lee distance d to that of a truncated distance ρ, defined in the next paragraph (not a standard distance, but see Remark 5) and subsequently applied in Section 2 that restates results of [12]). Furthermore, new graphs Γ n built as compounds of ternary n-cubes, presented below in Remark 1 and Sections 3-4 that generalize the lattice graph of Z n , extend those applications.…”

“…Also in the center-left of Figure 1, light-gray 3parallelotopes represent convex hull parts of truncated 3-spheres whose central vertices form a lattice PTDC, the simplest case of Theorem 2. A coloring-graph viewpoint of these constructions is found in [12].…”