The basis number of 1-planar graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866915078652035072 |
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| author | Bazargani, Saman Biedl, Therese Bose, Prosenjit Maheshwari, Anil Miraftab, Babak |
| author_facet | Bazargani, Saman Biedl, Therese Bose, Prosenjit Maheshwari, Anil Miraftab, Babak |
| contents | Let $B$ be a set of Eulerian subgraphs of a graph $G$. We say $B$ forms a $k$-basis if it is a minimum set that generates the cycle space of $G$, and any edge of $G$ lies in at most $k$ members of $B$. The basis number of a graph $G$, denoted by $b(G)$, is the smallest integer such that $G$ has a $k$-basis. A graph is called 1-planar (resp. planar) if it can be embedded in the plane with at most one crossing (resp. no crossing) per edge. MacLane's planarity criterion characterizes planar graphs based on their cycle space, stating that a graph is planar if and only if it has a $2$-basis. We study here the basis number of 1-planar graphs, demonstrate that it is unbounded in general, and show that it is bounded for many subclasses of 1-planar graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The basis number of 1-planar graphs Bazargani, Saman Biedl, Therese Bose, Prosenjit Maheshwari, Anil Miraftab, Babak Combinatorics Discrete Mathematics 05C10, 05C38, 05C76 Let $B$ be a set of Eulerian subgraphs of a graph $G$. We say $B$ forms a $k$-basis if it is a minimum set that generates the cycle space of $G$, and any edge of $G$ lies in at most $k$ members of $B$. The basis number of a graph $G$, denoted by $b(G)$, is the smallest integer such that $G$ has a $k$-basis. A graph is called 1-planar (resp. planar) if it can be embedded in the plane with at most one crossing (resp. no crossing) per edge. MacLane's planarity criterion characterizes planar graphs based on their cycle space, stating that a graph is planar if and only if it has a $2$-basis. We study here the basis number of 1-planar graphs, demonstrate that it is unbounded in general, and show that it is bounded for many subclasses of 1-planar graphs. |
| title | The basis number of 1-planar graphs |
| topic | Combinatorics Discrete Mathematics 05C10, 05C38, 05C76 |
| url | https://arxiv.org/abs/2412.18595 |