Spanning $H$-subdivisions and perfect $H$-subdivision tilings in dense digraphs
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| Formato: | Preprint |
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2024
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| _version_ | 1866914436153868288 |
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| author | Cheng, Yangyang Wang, Zhilan Yan, Jin |
| author_facet | Cheng, Yangyang Wang, Zhilan Yan, Jin |
| contents | Given a (di)graph $H$, we say that a (di)graph $H^\prime$ is an $H$-subdivision if $H^\prime$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Signé conjectured that for every $\varepsilon>0$, there exists a constant $C_0>0$ such that for every graph $H$ with $h$ edges and no isolated vertices, if $G$ is a graph on $n\geq C_0h$ vertices and minimum degree $δ(G)\geq(1+\varepsilon)\frac{n}{2}$, then $G$ contains a spanning $H$-subdivision. This conjecture was later resolved by Lee [European J. Combin. \textbf{124} (2025), 104059]. In this paper, we strengthen Lee's result. Specifically, we prove that for any digraph $D$ on $n\geq C_0h$ vertices, if the minimum semi-degree of $D$ is at least $\frac{n+h}{2}-1$, then $D$ contains a spanning $H$-subdivision. The lower bound on the minimum semi-degree is best possible.
Furthermore, we show that there exist constants $C>0$ and $α, β\in(0, 1)$ such that for any integer partition $n=n_1+\cdots+n_m\geq Cm$ with $n_i\geq|V(H)|+3h$ for each $i$, and $\sum_{n_i<αn}n_i\leqβn$, if a digraph of order $n\geq Cm$ has minimum semi-degree at least $\frac{n+m+h}{2}-1$, then it contains $m$ vertex-disjoint $H$-subdivisions whose orders are $n_1, \ldots, n_m$, respectively. The bound $\frac{n+m+h}{2}-1$ is also optimal. This work partly answers a conjecture of Lee [Combin. Probab. Comput. \textbf{34} (2025), 421--444] and generalizes a recent result from the same paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07786 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spanning $H$-subdivisions and perfect $H$-subdivision tilings in dense digraphs Cheng, Yangyang Wang, Zhilan Yan, Jin Combinatorics 05C20, 05C70, 05C07 Given a (di)graph $H$, we say that a (di)graph $H^\prime$ is an $H$-subdivision if $H^\prime$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Signé conjectured that for every $\varepsilon>0$, there exists a constant $C_0>0$ such that for every graph $H$ with $h$ edges and no isolated vertices, if $G$ is a graph on $n\geq C_0h$ vertices and minimum degree $δ(G)\geq(1+\varepsilon)\frac{n}{2}$, then $G$ contains a spanning $H$-subdivision. This conjecture was later resolved by Lee [European J. Combin. \textbf{124} (2025), 104059]. In this paper, we strengthen Lee's result. Specifically, we prove that for any digraph $D$ on $n\geq C_0h$ vertices, if the minimum semi-degree of $D$ is at least $\frac{n+h}{2}-1$, then $D$ contains a spanning $H$-subdivision. The lower bound on the minimum semi-degree is best possible. Furthermore, we show that there exist constants $C>0$ and $α, β\in(0, 1)$ such that for any integer partition $n=n_1+\cdots+n_m\geq Cm$ with $n_i\geq|V(H)|+3h$ for each $i$, and $\sum_{n_i<αn}n_i\leqβn$, if a digraph of order $n\geq Cm$ has minimum semi-degree at least $\frac{n+m+h}{2}-1$, then it contains $m$ vertex-disjoint $H$-subdivisions whose orders are $n_1, \ldots, n_m$, respectively. The bound $\frac{n+m+h}{2}-1$ is also optimal. This work partly answers a conjecture of Lee [Combin. Probab. Comput. \textbf{34} (2025), 421--444] and generalizes a recent result from the same paper. |
| title | Spanning $H$-subdivisions and perfect $H$-subdivision tilings in dense digraphs |
| topic | Combinatorics 05C20, 05C70, 05C07 |
| url | https://arxiv.org/abs/2411.07786 |