Minimal coloring number for Z-colorable links II

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The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link. In this paper, we show that the minimal coloring number of any non-splittable Z-colorable links is four. As an example, we consider the link obtained by replacing each component of the given link with several parallel strands, which we call a parallel of a link. We show that an even parallel of a link is Z-colorable except for the case of 2 parallels with non-zero linking number. We then give a simple way to obtain a diagram which attains the minimal coloring number for such even parallels of links.