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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2305.10008 |
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| _version_ | 1866912815249358848 |
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| author | Lv, Shengxiang |
| author_facet | Lv, Shengxiang |
| contents | The nearly complete bipartite graph $G(m,n,k)$ is obtained by removing $k$ independent edges from the complete bipartite graph $K_{m,n}$. In this paper, we prove that for any nearly complete bipartite graph $G(m,n,k)$ with $m, n\geq 3$, and $(m,n,k)\notin\{(5,4,4)$, $(4,5,4)$, $(5,5,5)\}$, there exists a nonorientable genus embedding $Π$ satisfying $\tildeγ(Π)=\max\{\lceil \big((m-2)(n-2)-k\big)/2\rceil, 1\}$. This embedding can be constructed by starting from an embedding of some $G(p,q,h)$ with $h\leq 6$ and $p,q\leq 7$, and then iteratively adding multiple copies of $G(2,2,2)$, $G(2,0,0)$ and $G(0,2,0)$. As a consequence, the previously unresolved nonorientable genus $\tildeγ(G(n+1,n,n))$ for even $n$ and $\tildeγ(G(n,n,n))$ for arbitrary $n$ are now determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_10008 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonorientable genus embedding of nearly complete bipartite graphs Lv, Shengxiang Combinatorics The nearly complete bipartite graph $G(m,n,k)$ is obtained by removing $k$ independent edges from the complete bipartite graph $K_{m,n}$. In this paper, we prove that for any nearly complete bipartite graph $G(m,n,k)$ with $m, n\geq 3$, and $(m,n,k)\notin\{(5,4,4)$, $(4,5,4)$, $(5,5,5)\}$, there exists a nonorientable genus embedding $Π$ satisfying $\tildeγ(Π)=\max\{\lceil \big((m-2)(n-2)-k\big)/2\rceil, 1\}$. This embedding can be constructed by starting from an embedding of some $G(p,q,h)$ with $h\leq 6$ and $p,q\leq 7$, and then iteratively adding multiple copies of $G(2,2,2)$, $G(2,0,0)$ and $G(0,2,0)$. As a consequence, the previously unresolved nonorientable genus $\tildeγ(G(n+1,n,n))$ for even $n$ and $\tildeγ(G(n,n,n))$ for arbitrary $n$ are now determined. |
| title | Nonorientable genus embedding of nearly complete bipartite graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.10008 |