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

TL;DR

This paper establishes a new upper bound for the DP-chromatic number of graph joins, improving previous bounds and providing insights into the relationship between chromatic and DP-chromatic numbers.

Contribution

The paper proves that for any graph, the DP-chromatic number of its join with a complete graph is equal to its chromatic number under certain conditions, leading to a tighter upper bound on Z_{DP}(n).

Findings

Z_{DP}(n)  n^2 - (n+3)/2 for all n  2.

Improves the upper bound from 1.5n^2 to n^2 - (n+3)/2.

Shows the equality   for joins with K_s under specified s.

Abstract

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 $\chi_{DP}(G \vee K_s)=\chi(G \vee K_s)$ holds for $s=\left \lceil \frac{4(k+1)m}{2k+1} \right \rceil \le \lceil 2.4m\rceil$, where $k=\chi(G)$, $m=|E(G)|$ and $G \vee K_s$ is the join of $G$ and the complete graph $K_s$. 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.