Rainbow domination numbers of hypercubes from covering-code tables
Year: 2026
Type: preprint
Abstract: The k-rainbow domination number γ_rk(G) has been determined for many families of graphs, such as grids, products of cycles and generalized Petersen graphs, but little is known for hypercubes. We point out that, by a theorem of Brešar, Henning and Rall, γ_r3(Q_n) is the size of an optimal mixed covering code with one ternary and n binary coordinates and radius 1, and, as an immediate observation, γ_r2(Q_n) is the size of an optimal binary covering code of length n+1 and radius 1. Both are tabulated in coding theory. This gives the exact values γ_r2(Q_n) for n ≤ 8 and γ_r3(Q_n) for n ≤ 6, and bounds beyond, which we confirm independently for small n by solving the rainbow domination problem directly. For the first open case, γ_r3(Q_7), for which the tables give 42 ≤ γ_r3(Q_7) ≤ 48, we show by computer that every 3-rainbow dominating function of Q_7 with weight at most 47 has no non-trivial symmetry.
Author: Parsons, Noah
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225598762