Cycles in graphs and in hypergraphs: results and problems

TL;DR

This paper explores the properties and enumeration of 1-cycles in graphs and hypergraphs, discussing their algebraic structure, counting problems, and generalizations involving symmetry and higher-dimensional cycles.

Contribution

It provides an expository overview of 1-cycle enumeration, their algebraic properties, and extends the discussion to hypergraphs and symmetric cases, highlighting open problems.

Findings

Characterization of 1-cycles in graphs

Methods for counting all 1-cycles

Extensions to hypergraphs and symmetric cases

Abstract

This is an expository paper. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. E.g., a cycle in the sense of graph theory is a $1$-cycle, but not vice versa. It is easy to check that the sum (modulo $2$) of $1$-cycles is a $1$-cycle. In this text we study the following problems: to find $\bullet$ the number of all 1-cycles in a given graph; $\bullet$ a small number of 1-cycles in a given graph such that any 1-cycle is the sum of some of them. We also consider generalizations (of these problems) to graphs with symmetry, and to $2$-cycles in $2$-dimensional hypergraphs.