Spectral radius and rainbow $k$-factors of graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913978284769280 |
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| author | Zhang, Liwen Zhang, Zhiyuan |
| author_facet | Zhang, Liwen Zhang, Zhiyuan |
| contents | Let $\mathcal{G}=\{G_1,\ldots, G_{\frac{kn}{2}}\}$ be a set of graphs on the same vertex set $V=\{1,\dots,n\}$ where $k\cdot n$ is even. We say $\mathcal{G}$ admits a rainbow $k$-factor if there exists a $k$-regular graph $F$ on the vertex set $V$ such that all edges of $F$ are from different members of $\mathcal{G}$. In this paper, we show a sufficient spectral condition for the existence of a rainbow $k$-factor for $k\geq 2$, which is that if $ρ(G_i)\geqρ(K_{k-1}\vee(K_1\cup K_{n-k}))$ for each $G_i\in \mathcal{G}$, then $\mathcal{G}$ admits a rainbow $k$-factor unless $G_1=G_2=\cdots=G_{\frac{kn}{2}}\cong K_{k-1}\vee(K_1\cup K_{n-k})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08162 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral radius and rainbow $k$-factors of graphs Zhang, Liwen Zhang, Zhiyuan Combinatorics Let $\mathcal{G}=\{G_1,\ldots, G_{\frac{kn}{2}}\}$ be a set of graphs on the same vertex set $V=\{1,\dots,n\}$ where $k\cdot n$ is even. We say $\mathcal{G}$ admits a rainbow $k$-factor if there exists a $k$-regular graph $F$ on the vertex set $V$ such that all edges of $F$ are from different members of $\mathcal{G}$. In this paper, we show a sufficient spectral condition for the existence of a rainbow $k$-factor for $k\geq 2$, which is that if $ρ(G_i)\geqρ(K_{k-1}\vee(K_1\cup K_{n-k}))$ for each $G_i\in \mathcal{G}$, then $\mathcal{G}$ admits a rainbow $k$-factor unless $G_1=G_2=\cdots=G_{\frac{kn}{2}}\cong K_{k-1}\vee(K_1\cup K_{n-k})$. |
| title | Spectral radius and rainbow $k$-factors of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.08162 |