On the Minimal Sum of Edges in a Signed Edge-Dominated Graph

Abstract

Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to $e$ (including $e$ itself) is positive. Then the sum of weights over edges of $G$ is at least $-\frac{n^2}{25}$. Also we provide an example of a weighted graph with described properties and the sum of weights $-(1+o(1))\frac{n^2}{8(1 + \sqrt{2})^2}$.

The previous best known bounds were $-\frac{n^2}{16}$ and $-(1+o(1))\frac{n^2}{54}$ respectively.
We show that the constant $-1/54$ is optimal under some additional conditions.