on n vertices with minimum degree r, there exists a two-coloring of the vertices of G with colors +1 and -1, such that the closed neighborhood of each vertex contains more +1's than -1's, and altogether the number of 1's does not exceed the number of -1's by more than \(\). As a construction by Füredi and Mubayi shows, this is asymptotically tight. The proof uses the partial coloring method from combinatorial discrepancy theory.
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