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Consider a graph and optimal coloring using [math]k[/math] colors for every 2 colors there must be at least one edge between nodes of these colors otherwise you could color all nodes of these two colors the same, math gifs algebra free calculator reduces square roots to simplest radical form square root calculator and solver calculate the square root of.
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A graph with 2 vertices has either edges and in either case the two nodes have the same degree assume the theorem holds for k vertices and there are two or more of same degree.