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It was shown that any $\mathbb{Z}$-colorable link has a diagram which admitsa non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, weconsider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings onminimal diagrams of $\mathbb{Z}$-colorable links. We show, for any positiveinteger $N$, there exists a minimal diagram of a $\mathbb{Z}$-colorable linksuch that any $\mathbb{Z}$-coloring on the diagram has at least $N$ colors. Onthe other hand, it is shown that certain $\mathbb{Z}$-colorable torus linkshave minimal diagrams admitting $\mathbb{Z}$-colorings with only four colors.

Minimal coloring number on minimal diagrams for $\mathbb{Z}$-colorable links