Cycles in graphs and in hypergraphs

08/09/2023

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by   E. Alkin, et al.

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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 ∙ the number of all 1-cycles in a given graph; ∙ 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.