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.

翻译：这是一篇阐述性论文。图中的一个$1$-环是指边集$C$，使得每个顶点包含在$C$中偶数条边上。例如，图论意义下的环是一个$1$-环，但反之不成立。易证$1$-环的模$2$和仍为$1$-环。本文研究以下问题：找出$\bullet$给定图中所有$1$-环的数量；$\bullet$给定图中少量$1$-环，使得任意$1$-环均为其中若干环的和。我们同时考虑这些问题的推广形式，包括对称图的情形以及二维超图中的$2$-环情形。