On the Signed Domination in Graphs

We prove a conjecture of F uredi and Mubayi: For any graph 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 O(n= p r). As a construction by F uredi and Mubayi shows, this is asymptotically tight. The proof uses the partial coloring method from combinatorial discrepancy theory. Let G be a (simple, undirected) graph on n vertices. For a vertex v 2 V (G), the closed neighborhood Nv] of v is the set consisting of v and all of its neighbors. A signed domination function of G is any function : V (G) ! f?1; +1g such that for every vertex v 2 V (G), we have (Nv]) > 0 (here and in the sequel, we use the notation (S) = P x2S (x) for a subset S of the domain of). The signed domination number of G, s (G), is deened as s (G) = minf(V (G)) : is a signed domination function of Gg: This variant of the usual domination number was introduced by Dunbar et al. 4] in the early 1990s. Several researchers have studied estimates for the largest possible value of s (G) for r-regular n-vertex graphs (or for n-vertex graphs of minimum degree r) in dependence on n and on r (see 5] for references). Recently F uredi and Mubayi 5] proved, by a simple probabilistic argument, that for any n-vertex graph G of minimum degree r, s (G) 2 q log r r + 1 r n holds. They also constructed an r-regular graph on 4r vertices with s (G) 1 2 p r ? O(1), which shows that the upper bound is asymptotically nearly tight, up to the factor of p log r. They conjectured that the lower bound is in fact asymptotically optimal, i.e. that all n-vertex graphs of minimum degree r have signed domination number O(n= p r). For the special case of r-regular n-vertex graphs, they derived this conjecture from a long-standing conjecture of Beck and Fiala 3] in discrepancy theory. Here we prove the F uredi{Mubayi conjecture, using the so-called partial coloring method from combinatorial discrepancy theory (invented by Beck 2] and reened by Spencer 7]). …