On the Signed Domination in Graphs

G

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.