Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2402.05458 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917585042276352 |
|---|---|
| author | Gao, Hui |
| author_facet | Gao, Hui |
| contents | In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences, submitted], which characterizes a mixed hypergraph $\mathcal{F}=(V, \mathcal{E} \cup \mathcal{A})$ having an orientation $\overrightarrow{\mathcal{E}}$ of $\mathcal{E}$ such that $e_{\overrightarrow{\mathcal{E}} \cup \mathcal{A}} (\mathcal{P}) \geq \sum_{X \in \mathcal{P}}h(X) -b(\cup \mathcal{P})$ for every subpartition $\mathcal{P}$ of $V$, where $h$ is an integer-valued, intersecting supermodular function on $V$ and $b$ a submodular function on $V$. As a corollary, another conjecture in the same paper is confirmed, which characterizes a mixed hypergraph having a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex $v$ belongs to exactly $k$ of them and is the root of at least $f(v)$ and at most $g(v)$ of them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05458 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Covering a supermodular-like function in a mixed hypergraph Gao, Hui Combinatorics In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences, submitted], which characterizes a mixed hypergraph $\mathcal{F}=(V, \mathcal{E} \cup \mathcal{A})$ having an orientation $\overrightarrow{\mathcal{E}}$ of $\mathcal{E}$ such that $e_{\overrightarrow{\mathcal{E}} \cup \mathcal{A}} (\mathcal{P}) \geq \sum_{X \in \mathcal{P}}h(X) -b(\cup \mathcal{P})$ for every subpartition $\mathcal{P}$ of $V$, where $h$ is an integer-valued, intersecting supermodular function on $V$ and $b$ a submodular function on $V$. As a corollary, another conjecture in the same paper is confirmed, which characterizes a mixed hypergraph having a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex $v$ belongs to exactly $k$ of them and is the root of at least $f(v)$ and at most $g(v)$ of them. |
| title | Covering a supermodular-like function in a mixed hypergraph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.05458 |