k-Domination ivariants on Kneser graphs

• Autores: Brešar, Boštjan; Dravec, Tanja; Cornet, María Gracia; Henning, Michael A.
• Año de publicación: 2025
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Kneser graphs; K-domination; K-tuple total domination; 2-packing
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

In this follow-up to work of M.G. Cornet and P. Torres from 2023, where the k-tuple domination number and the 2-packing number in Kneser graphs K(n, r) were studied, we are concerned with two variations, the k-domination number, γk(K(n, r)), and the k-tuple total domination number, γt × k(K(n, r)), of K(n, r). For both invariants we prove monotonicity results by showing that γk(K(n, r)) ≥ γk(K(n + 1, r)) holds for any n ≥ 2(k + r), and γt × k(K(n, r)) ≥ γt × k(K(n + 1, r)) holds for any n ≥ 2r + 1. We prove that γk(K(n, r)) = γt × k(K(n, r)) = k + r when n ≥ r(k + r), and that in this case every γ_(k)-set and γ_(t × k)-set is a clique, while γk(r(k + r) − 1, r) = γt × k(r(k + r) − 1, r) = k + r + 1, for any k ≥ 2. Concerning the 2-packing number, ρ₂(K(n, r)), of K(n, r), we prove the exact values of ρ₂(K(3r − 3, r)) when r ≥ 10, and give sufficient conditions for ρ₂(K(n, r)) to be equal to some small values by imposing bounds on r with respect to n. We also prove a version of monotonicity for the 2-packing number of Kneser graphs.
Fil: Brešar, Boštjan. University Of Maribor; Eslovenia
Fil: Dravec, Tanja. University Of Maribor; Eslovenia
Fil: Cornet, María Gracia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Henning, Michael A.. University Of Johannesburg; Sudáfrica