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K. Ichihara and E. Matsudo introduced the notions of Z-colorable links and the minimal coloring number for Z-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable Z-colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable Z-colorable link is exactly 4.

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