A New Lower Bound on the Potential-Ramsey Number of Two Graphs

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 a graph H, a graphic sequence π is potentially H-graphic if π has a realization containing H as a subgraph. For graphs G₁ and G₂, the potential-Ramsey number r_pot(G₁, G₂) is the smallest integer k such that for every k-term graphic sequence π, either π is potentially G₁-graphic or the complementary sequence $\overline \pi = \left({k - 1 - {d_k}, \cdots k - 1 - {d_1}} \right)$ is potentially G₂-graphic. For $0 \le k \le \left\lfloor {{t \over 2}} \right\rfloor$, denote $K_t^{- k}$ to be the graph obtained from K_t by deleting k independent edges. If k = 0, Busch et al. (Graphs Combin., 30(2014)847–859) present a lower bound on r_pot(G, K_t) by using the 1-dependence number of G. In this paper, we utilize i-dependence number of G for i ≥ 1 to give a new lower bound on r_pot(G, $K_t^{- k}$) for any k with $0 \le k \le \left\lfloor {{t \over 2}} \right\rfloor$. Moreover, we also determine the exact values of r_pot(K_n, $K_t^{- k}$) for 1 ≤ k ≤ 2.