# Negative (and Positive) Circles in Signed Graphs: A Problem Collection

## TL;DR

This paper presents a collection of open problems and questions related to negative and positive cycles in signed graphs, exploring their properties, counts, packings, coverings, and spectral aspects.

## Contribution

It compiles and discusses various research questions on signed circles, highlighting areas for future investigation and providing some partial answers.

## Key findings

- Some questions about signed circles have been answered.
- The paper identifies key problems in counting and characterizing signed cycles.
- Connections between signed cycles and eigenvalues are explored.

## Abstract

A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. Most of the essential properties of a signed graph depend on the signs of its circles. Here I describe several questions regarding negative circles and their cousins the positive circles. Topics include incidence between signed circles and edges or vertices, characterizing signed graphs with special circle properties, counting negative circles, signed-circle packing and covering, signed circles and eigenvalues, and directed cycles in signed digraphs. A few of the questions come with answers.