Title: Minimal coloring number on minimal diagrams for $\mathbb{Z}$-colorable links,
Abstract: It was shown that any $\mathbb{Z}$-colorable link has a diagram which admits
a non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, we
consider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings on
minimal diagrams of $\mathbb{Z}$-colorable links. We show, for any positive
integer $N$, there exists a minimal diagram of a $\mathbb{Z}$-colorable link
such that any $\mathbb{Z}$-coloring on the diagram has at least $N$ colors. On
the other hand, it is shown that certain $\mathbb{Z}$-colorable torus links
have minimal diagrams admitting $\mathbb{Z}$-colorings with only four colors.