### A further result on the potential-Ramsey number of $G_1$ and $G_2$

#### Abstract

A non-increasing sequence

$\pi=(d_1,\ldots,d_n)$ of nonnegative integers is a {\it graphic

sequence} if it is realizable by a simple graph $G$ on $n$ vertices.

In this case, $G$ is referred to as a {\it realization} of $\pi$.

Given a graph $H$, a graphic sequence $\pi$ is {\it potentially

$H$-graphic} if $\pi$ has a realization containing $H$ as a

subgraph. Busch et al. (Graphs Combin., 30(2014)847--859) considered

a degree sequence analogue to classical graph Ramsey number as

follows: for graphs $G_1$ and $G_2$, the {\it potential-Ramsey

number} $r_{pot}(G_1,G_2)$ is the smallest non-negative integer $k$

such that for any $k$-term graphic sequence $\pi$, either $\pi$ is

potentially $G_1$-graphic or the complementary sequence

$\overline{\pi}=(k-1-d_k,\ldots,k-1-d_1)$ is potentially

$G_2$-graphic. They also gave a lower bound on $r_{pot}(G,K_{r+1})$

for a number of choices of $G$ and determined the exact values for

$r_{pot}(K_n,K_{r+1})$, $r_{pot}(C_n,K_{r+1})$ and

$r_{pot}(P_n,K_{r+1})$. In this paper, we will extend the complete

graph $K_{r+1}$ to the complete split graph

$S_{r,s}=K_r\vee\overline{K_s}$. Clearly, $S_{r,1}=K_{r+1}$. We

first give a lower bound on $r_{pot}(G,S_{r,s})$ for a number of

choices of $G$, and then determine the exact values for

$r_{pot}(C_n,S_{r,s})$ and $r_{pot}(P_n,S_{r,s})$.

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