The potential-Ramsey numbers rpot(Cn,Kt−k) and rpot(Pn,Kt−k)

A nonincreasing sequence ρ=(ρ1,…,ρn) 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 G1 and G2, Busch et al. introduced the potential-Ramsey number of G1 and G2, denoted rpot(G1,G2), is the smallest nonnegative integer m such that for every m-term graphic sequence ρ, there is a realization G of ρ with G1⊆G or with G2⊆G¯, where G¯ is the complement of G. For t≥2 and 0≤k≤⌊t2⌋, let Kt−k be the graph obtained from Kt by deleting k independent edges. Busch et al. determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=0. Du and Yin determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=1. In this paper, we further determine rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for 2≤k≤⌊t2⌋.