A vector k-coloring of G is an assignment i Î N ® ui Î Rn such that ui uj= 1 and ui uj  £ -1 /(k - 1) if (i, j) Î E (k need not be an integer). Note that for k = 2 only two opposite vectors can be assigned, thus a vector 2-coloring is identical to a normal 2-coloring.In general a k-coloring implies a vector k-coloring but not viceversa. To prove it consider the following vectors (exemplified for k = 3 and n = 5).

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