<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def color_mapping(sample_map, header, group_column, color_column=None):
 """ Determine color-category mapping. If color_column was specified, then map the category names to color values. Otherwise, use the palettable colors to automatically generate a set of colors for the group values. :type sample_map: dict :param unifracFN: Map associating each line of the mapping file with the appropriate sample ID (each value of the map also contains the sample ID) :type header: tuple :param A tuple of header line for mapping file :type group_column: str :param group_column: String denoting the column name for sample groups. :type color_column: str :param color_column: String denoting the column name for sample colors. :type return: dict :param return: {SampleID: Color} """

    group_colors = OrderedDict()
    group_gather = gather_categories(sample_map, header, [group_column])

    if color_column is not None:
        color_gather = gather_categories(sample_map, header, [color_column])
        # match sample IDs between color_gather and group_gather
        for group in group_gather:
            for color in color_gather:
                # allow incomplete assignment of colors, if group sids overlap at
                # all with the color sids, consider it a match
                if group_gather[group].sids.intersection(color_gather[color].sids):
                    group_colors[group] = color
    else:
        bcolors = itertools.cycle(Set3_12.hex_colors)
        for group in group_gather:
            group_colors[group] = bcolors.next()

    return group_colors