The potential-Ramsey numbers r p o t ( C n , K t − k ) and r p o t ( P n , K t − k )

• Determine the value r p o t ( P n , K t − k ) for 2 ≤ k ≤ ⌊ t 2 ⌋. • Determine the value r p o t ( C n , K t − k ) for 2 ≤ k ≤ ⌊ t 2 ⌋. A nonincreasing sequence ρ = ( ρ 1 , … , ρ n ) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ρ. Given two graphs G 1 and G 2, Busch et al. introduced the potential-Ramsey number of G 1 and G 2, denoted r p o t ( G 1 , G 2 ), is the smallest nonnegative integer m such that for every m-term graphic sequence ρ, there is a realization G of ρ with G 1 ⊆ G or with G 2 ⊆ G ¯, where G ¯ is the complement of G. For t ≥ 2 and 0 ≤ k ≤ ⌊ t 2 ⌋, let K t − k be the graph obtained from K t by deleting k independent edges. Busch et al. determined r p o t ( C n , K t − k ) and r p o t ( P n , K t − k ) for k = 0. Du and Yin determined r p o t ( C n , K t − k ) and r p o t ( P n , K t − k ) for k = 1. In this paper, we further determine r p o t ( C n , K t − k ) and r p o t ( P n , K t − k ) for 2 ≤ k ≤ ⌊ t 2 ⌋.