Rainbow perfect domination in lattice graphs

• Main Authors: Luis R. Fuentes, Italo J. Dejter, Carlos A. Araujo
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2018-04-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: perfect dominating set, lattice, tiling, rainbow sphere
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/265

<p>Let <span class="math">0 &lt; <em>n</em> ∈ Z</span>. In the unit distance graph of <span class="math">Z^<em>n</em> ⊂ R^<em>n</em></span>, 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 <span class="math"><em>C</em></span> having: <span>i</span> vertex sets <span class="math"><em>V</em>(<em>C</em>)</span> whose convex hulls are n-parallelotopes (resp., both <span class="math">(<em>n</em> − 1</span>)- and 0-cubes) and <span>ii</span> each <span class="math"><em>V</em>(<em>C</em>)</span> contained in a corresponding rainbow sphere centered at <span class="math"><em>C</em></span> with radius <span class="math"><em>n</em></span> (resp., radii 1 and <span class="math"><em>n</em> − 2</span>).</p>