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I have a layer in QGIS showing U.S. counties. I would like to color-fill the polygons in a layer using a range of 5 colors, so that no colors are adjacent to each other.

Is this possible?

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QGIS 3.0 has this ability : just check topological coloring in the toolbox processes (in "cartography") ... (u can even choose the number of colours to use - a minimum is automatically computed)

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In the end, I couldn't get this working with QGIS 2.18, so used an old copy of MapInfo with a plugin called RouteWare ToolBox. It works brilliantly, and adds an integer field to the shapefile with numbers that can then be assigned to a table of colors.

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