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The minimal coloring number of a $\mathbb{Z}$-colorable link is the minimalnumber of colors for non-trivial $\mathbb{Z}$-colorings on diagrams of thelink. In this paper, we show that the minimal coloring number of anynon-splittable $\mathbb{Z}$-colorable links is four. As an example, we considerthe link obtained by replacing each component of the given link with severalparallel strands, which we call a parallel of a link. We show that an evenparallel of a link is $\mathbb{Z}$-colorable except for the case of 2 parallelswith non-zero linking number. We then give a simple way to obtain a diagramwhich attains the minimal coloring number for such even parallels of links.

Minimal coloring number for $\mathbb{Z}$-colorable links II