$Z_{DP}(n)$ is upperly bounded by $n^2-(n+3)\mathrm{/}2$

DP-coloring was introduced by Dvo\v{r}'{a}k and Postle and is a generalization of proper coloring. For any graph $G$, let $\chi (G)$ and ${\chi }_{DP}(G)$ denote the chromatic number and the DP-chromatic number of $G$ respectively. In this article, we show that $\chi_{DP}(G \vee K_s)=\chi(G \vee K_s)$holdsfor$s=\left \lceil \frac{4(k+1)m}{2k+1} \right \rceil \le \lceil 2.4m\rceil$,where$k=\chi(G)$,$m=|E(G)|$and$G \vee K_s$isthejoinof$G$andthecompletegraph$K_s$\mathrm{.}Hence$Z_{DP}(n)\le n^2-(n+3)/2$ holds for every integer $n\ge 2$, where $Z_{DP}(n)$ is the minimum natural number $s$ such that ${\chi }_{DP}(G\vee K_s)=\chi (G\vee K_s)$ holds for every graph $G$ of order $n$. Our result improves the best current upper bound $Z_{DP}(n)\le 1.5n^2$ due to Bernshteyn, Kostochka and Zhu.