k-Domination invariants on Kneser graphs

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, γ_k(K(n, r)), and the ktuple total domination number, γt×_k(K(n, r)), of K(n, r). For both invariants we prove monotonicity results by showing that γ_k(K(n, r)) ≥ γ_k(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 γ_k(K(n, r)) = γt×_k(K(n, r)) = k + r when n ≥ r(k + r), and that in this case every γkset and γt×_k-set is a clique, while γ_k(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.