Negative Circles in Signed Graphs: A Problem Collection

TL;DR

This paper explores various fundamental problems related to positive and negative cycles in signed graphs, highlighting their importance and potential for advancing graph theory.

Contribution

It introduces a collection of natural and significant open problems about circles in signed graphs, connecting classical cycle problems to the signed graph setting.

Findings

Identifies key problems about positive and negative cycles in signed graphs.

Highlights the importance of signed graph theory in understanding cycle problems.

Proposes a research agenda for future work in this area.

Abstract

I propose that most problems about circles (cycles, circuits) in ordinary graphs that have odd or even length find their proper setting in the theory of signed graphs, where each edge has a sign, $+$ or $-$. Even-circle and odd-circle problems correspond to questions about positive and negative circles in signed graphs. (The sign of a circle is the product of its edge signs.) I outline questions about circles in signed graphs, that seem natural and potentially important.