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| Autores principales: | , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2408.08598 |
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| _version_ | 1866914913706835968 |
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| author | Buchanan, Calum Clifton, Alexander Culver, Eric Frankl, Péter Nie, Jiaxi Ozeki, Kenta Rombach, Puck Yin, Mei |
| author_facet | Buchanan, Calum Clifton, Alexander Culver, Eric Frankl, Péter Nie, Jiaxi Ozeki, Kenta Rombach, Puck Yin, Mei |
| contents | Babai and Frankl posed the ``odd cover problem" of finding the minimum cardinality of a collection of complete bipartite graphs such that every edge of the complete graph of order $n$ is covered an odd number of times. In a previous paper with O'Neill, some of the authors proved that this value is always $\lceil n / 2 \rceil$ or $\lceil n / 2 \rceil + 1$ and that it is the former whenever $n$ is a multiple of $8$. In this paper, we determine this value to be $\lceil n / 2 \rceil$ whenever $n$ is odd or equivalent to $18$ modulo $24$. We also further the study of odd covers of graphs which are not complete, wherein edges are covered an odd number of times and nonedges an even number of times by the complete bipartite graphs in the collection. Among various results on disjoint unions, we find the minimum cardinality of an odd cover of a union of odd cliques and of a union of cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08598 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On odd covers of cliques and disjoint unions Buchanan, Calum Clifton, Alexander Culver, Eric Frankl, Péter Nie, Jiaxi Ozeki, Kenta Rombach, Puck Yin, Mei Combinatorics 05C70, 05C50 Babai and Frankl posed the ``odd cover problem" of finding the minimum cardinality of a collection of complete bipartite graphs such that every edge of the complete graph of order $n$ is covered an odd number of times. In a previous paper with O'Neill, some of the authors proved that this value is always $\lceil n / 2 \rceil$ or $\lceil n / 2 \rceil + 1$ and that it is the former whenever $n$ is a multiple of $8$. In this paper, we determine this value to be $\lceil n / 2 \rceil$ whenever $n$ is odd or equivalent to $18$ modulo $24$. We also further the study of odd covers of graphs which are not complete, wherein edges are covered an odd number of times and nonedges an even number of times by the complete bipartite graphs in the collection. Among various results on disjoint unions, we find the minimum cardinality of an odd cover of a union of odd cliques and of a union of cycles. |
| title | On odd covers of cliques and disjoint unions |
| topic | Combinatorics 05C70, 05C50 |
| url | https://arxiv.org/abs/2408.08598 |