On intersecting families of subgraphs of perfect matchings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911957856026624 |
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| author | Fuentes, Melissa M. Kamat, Vikram |
| author_facet | Fuentes, Melissa M. Kamat, Vikram |
| contents | The seminal Erdős--Ko--Rado (EKR) theorem states that if $\mathcal{F}$ is a family of $k$-subsets of an $n$-element set $X$ for $k\leq n/2$ such that every pair of subsets in $\mathcal{F}$ has a nonempty intersection, then $\mathcal{F}$ can be no bigger than the trivially intersecting family obtained by including all $k$-subsets of $X$ that contain a fixed element $x\in X$. This family is called the star centered at $x$. In this paper, we formulate and prove an EKR theorem for intersecting families of subgraphs of the perfect matching graph, the graph consisting of $n$ disjoint edges. This can be considered a generalization not only of the aforementioned EKR theorem but also of a signed variant of it, first stated by Meyer (1974), and proved separately by Deza--Frankl (1983) and Bollobás--Leader (1997). The proof of our main theorem relies on a novel extension of Katona's beautiful cycle method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On intersecting families of subgraphs of perfect matchings Fuentes, Melissa M. Kamat, Vikram Combinatorics 05D05 (Primary), 05C35 (Secondary) The seminal Erdős--Ko--Rado (EKR) theorem states that if $\mathcal{F}$ is a family of $k$-subsets of an $n$-element set $X$ for $k\leq n/2$ such that every pair of subsets in $\mathcal{F}$ has a nonempty intersection, then $\mathcal{F}$ can be no bigger than the trivially intersecting family obtained by including all $k$-subsets of $X$ that contain a fixed element $x\in X$. This family is called the star centered at $x$. In this paper, we formulate and prove an EKR theorem for intersecting families of subgraphs of the perfect matching graph, the graph consisting of $n$ disjoint edges. This can be considered a generalization not only of the aforementioned EKR theorem but also of a signed variant of it, first stated by Meyer (1974), and proved separately by Deza--Frankl (1983) and Bollobás--Leader (1997). The proof of our main theorem relies on a novel extension of Katona's beautiful cycle method. |
| title | On intersecting families of subgraphs of perfect matchings |
| topic | Combinatorics 05D05 (Primary), 05C35 (Secondary) |
| url | https://arxiv.org/abs/2407.12289 |