ZDP(n) ${Z}_{DP}(n)$ is bounded above by n2-(n+3)/2 ${n}∧{2}-(n+3)\unicode{x02215}2$

In 2018, Dvorak and Postle introduced a generalization of proper coloring, the so-called DP-coloring. For any graph G $G$, the DP-chromatic number chi DP(G) ${\chi }_{DP}(G)$ of G $G$ is defined analogously with the chromatic number chi(G) $\chi (G)$ of G $G$. In this article, we show that chi DP(G proves Ks)=chi(G proves Ks) ${\chi }_{DP}(G\vee {K}_{s})=\chi (G\vee {K}_{s})$ holds for s=4(chi(G)+1)|E(G)|2 chi(G)+1 $s=\unicode{x02308}\frac{4(\chi (G)+1)|E(G)|}{2\chi (G)+1}\unicode{x02309}$, where G proves Ks $G\vee {K}_{s}$ is the join of G $G$ and a complete graph with s $s$ vertices. As a result, ZDP(n)<= n2-(n+3)/2 ${Z}_{DP}(n)\le {n}<^>{2}-(n+3)\unicode{x02215}2$ holds for every integer n >= 2 $n\ge 2$, where ZDP(n) ${Z}_{DP}(n)$ is the minimum nonnegative integer s $s$ such that chi DP(G proves Ks)=chi(G proves Ks) ${\chi }_{DP}(G\vee {K}_{s})=\chi (G\vee {K}_{s})$ holds for every graph G $G$ with n $n$ vertices. Our result improves the best current upper bound 1.5n2 $1.5{n}<^>{2}$ of ZDP(n) ${Z}_{DP}(n)$ due to Bernshteyn, Kostochka, and Zhu.