The potential-Ramsey number of K_n and K ^−k_t

Abstract

A nonincreasing sequence π = (d₁,…, d_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 π. Given two graphs G₁ and G₂, A. Busch et al. (2014) introduced the potential-Ramsey number of G₁ and G₂, denoted by r_pot(G₁, G₂), as the smallest nonnegative integer m such that for every m-term graphic sequence π, there is a realization G of π with G₁ ⊆ G or with ${G_2} \subseteq \overline G$, where $\overline G$ is the complement of G. For t ≽ 2 and $0\leqslant k\leqslant \left\lfloor {{t \over 2}} \right\rfloor$, let K ^−k_t be the graph obtained from K_t by deleting k independent edges. We determine r_pot (K_n, K ^−k_t ) for $t\geqslant 3,1\leqslant k\leqslant \left\lfloor {{t \over 2}} \right\rfloor$ and $n\geqslant \left\lceil {\sqrt {2k}} \right\rceil$, which gives the complete solution to a result in J. Z. Du, J. H. Yin (2021).