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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.05477 |
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| _version_ | 1866910675447578624 |
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| author | Ferber, Asaf Han, Jie Mao, Dingjia |
| author_facet | Ferber, Asaf Han, Jie Mao, Dingjia |
| contents | Given a family of graphs $G_1,\dots,G_{n}$ on the same vertex set $[n]$, a rainbow Hamilton cycle is a Hamilton cycle on $[n]$ such that each $G_c$ contributes exactly one edge. We prove that if $G_1,\dots,G_{n}$ are independent samples of $G(n,p)$ on the same vertex set $[n]$, then for each $\varepsilon>0$, whp, every collection of spanning subgraphs $H_c\subseteq G_c$, with $δ(H_c)\geq(\frac{1}{2}+\varepsilon)np$, admits a rainbow Hamilton cycle. A similar result is proved for rainbow perfect matchings in a family of $n/2$ graphs on the same vertex set $[n]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05477 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Dirac-type Problem of Rainbow matchings and Hamilton cycles in Random Graphs Ferber, Asaf Han, Jie Mao, Dingjia Combinatorics Given a family of graphs $G_1,\dots,G_{n}$ on the same vertex set $[n]$, a rainbow Hamilton cycle is a Hamilton cycle on $[n]$ such that each $G_c$ contributes exactly one edge. We prove that if $G_1,\dots,G_{n}$ are independent samples of $G(n,p)$ on the same vertex set $[n]$, then for each $\varepsilon>0$, whp, every collection of spanning subgraphs $H_c\subseteq G_c$, with $δ(H_c)\geq(\frac{1}{2}+\varepsilon)np$, admits a rainbow Hamilton cycle. A similar result is proved for rainbow perfect matchings in a family of $n/2$ graphs on the same vertex set $[n]$. |
| title | Dirac-type Problem of Rainbow matchings and Hamilton cycles in Random Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.05477 |