On the minimal sum of edges in a signed edge-dominated graph

TL;DR

This paper investigates the minimal total edge weight in signed graphs where each edge's adjacent edges sum positively, establishing new bounds and demonstrating the optimality of certain constants under specific conditions.

Contribution

The paper introduces improved bounds on the minimal sum of edge weights in signed graphs with positive local sums and proves the optimality of a particular constant under additional constraints.

Findings

New lower bound of -n^2/25 for the sum of edge weights.

Constructed example with sum close to -n^2/8(1+√2)^2.

Demonstrated the optimality of -1/54 constant under extra conditions.

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.