Cycles in graphs and in hypergraphs: results and problems
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914882078638080 |
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| author | Alkin, E. Dzhenzher, S. Nikitenko, O. Skopenkov, A. Voropaev, A. |
| author_facet | Alkin, E. Dzhenzher, S. Nikitenko, O. Skopenkov, A. Voropaev, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05175 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cycles in graphs and in hypergraphs: results and problems Alkin, E. Dzhenzher, S. Nikitenko, O. Skopenkov, A. Voropaev, A. History and Overview Discrete Mathematics Algebraic Topology Combinatorics 55-01, 05-01, 05C25, 05C65 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. |
| title | Cycles in graphs and in hypergraphs: results and problems |
| topic | History and Overview Discrete Mathematics Algebraic Topology Combinatorics 55-01, 05-01, 05C25, 05C65 |
| url | https://arxiv.org/abs/2308.05175 |