Transversal tilings in k-partite graphs without large holes
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910018818801664 |
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| author | He, Xinyu Nie, Xiangxiang Yang, Donglei |
| author_facet | He, Xinyu Nie, Xiangxiang Yang, Donglei |
| contents | We show that for any constant $μ>0$ and $k\ge 3$, there exists $α>0$ such that the following holds for sufficiently large $n \in \mathbb{N}$. If $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $K_{k}$ with ${δ^*}(G)\geq (\frac{1}{2}+μ) n$ and $α^*_{k-1}(G)<αn$, then $G$ has a transversal $K_{k}$-factor. Moreover, the bound $\frac{1}{2}$ is asymptotically tight for the case \(k=3\). In addition, we show that if $k\ge 4$, $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $C_{k}$ with ${δ^*}(G)\ge (\frac{2}{k}+μ) n$, and $α^*_{2}(G)<αn$, then $G$ has a transversal $C_{k}$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transversal tilings in k-partite graphs without large holes He, Xinyu Nie, Xiangxiang Yang, Donglei Combinatorics We show that for any constant $μ>0$ and $k\ge 3$, there exists $α>0$ such that the following holds for sufficiently large $n \in \mathbb{N}$. If $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $K_{k}$ with ${δ^*}(G)\geq (\frac{1}{2}+μ) n$ and $α^*_{k-1}(G)<αn$, then $G$ has a transversal $K_{k}$-factor. Moreover, the bound $\frac{1}{2}$ is asymptotically tight for the case \(k=3\). In addition, we show that if $k\ge 4$, $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $C_{k}$ with ${δ^*}(G)\ge (\frac{2}{k}+μ) n$, and $α^*_{2}(G)<αn$, then $G$ has a transversal $C_{k}$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang. |
| title | Transversal tilings in k-partite graphs without large holes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.10578 |