Strong binding numbers and factors
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915463190020096 |
|---|---|
| author | Chen, Guantao Lavrov, Mikhail Ma, Yuying Vandenbussche, Jennifer van der Holst, Hein |
| author_facet | Chen, Guantao Lavrov, Mikhail Ma, Yuying Vandenbussche, Jennifer van der Holst, Hein |
| contents | Let $G$ be a simple graph. The $k$-th neighborhood of a vertex subset $S \subseteq V(G)$, denoted $Λ^k(S)$, is the set of vertices that are adjacent to at least $k$ vertices in $S$. The $k$-th binding number $β^k(G)$ is defined as the minimum ratio $|Λ^k(S)|/|S|$ over all subsets $S \subseteq V(G)$ with $|S| \ge k$ and $Λ^k(S) \ne V(G)$. This parameter generalizes the classical binding number introduced by Woodall. Andersen showed that the condition $β^1(G) \ge 1$ does not guarantee the existence of a $1$-factor in $G$, while Barát et al. proved that $β^2(G) \ge 1$ suffices for the existence of a $2$-factor. In this paper, we extend this result to general $k \ge 2$ by showing that any graph $G$ with even $k|V(G)|$ and $β^k(G) \ge 1$ contains a $k$-factor. Moreover, if $G$ is additionally a split graph of even order, then it admits a $(k+1)$-factor. We also prove that any graph $G$ with $β^k(G) \ge 1$ contains at least $k-1$ disjoint perfect or near-perfect matchings. Finally, for any bipartite graph $G$ with bipartition $(X, Y)$, we introduce an analogue of the $k$-th binding number and show that, under the condition $β^k(G, X) \ge 1$, the graph admits $k$ disjoint matchings, each covering $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18555 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong binding numbers and factors Chen, Guantao Lavrov, Mikhail Ma, Yuying Vandenbussche, Jennifer van der Holst, Hein Combinatorics 05C70 Let $G$ be a simple graph. The $k$-th neighborhood of a vertex subset $S \subseteq V(G)$, denoted $Λ^k(S)$, is the set of vertices that are adjacent to at least $k$ vertices in $S$. The $k$-th binding number $β^k(G)$ is defined as the minimum ratio $|Λ^k(S)|/|S|$ over all subsets $S \subseteq V(G)$ with $|S| \ge k$ and $Λ^k(S) \ne V(G)$. This parameter generalizes the classical binding number introduced by Woodall. Andersen showed that the condition $β^1(G) \ge 1$ does not guarantee the existence of a $1$-factor in $G$, while Barát et al. proved that $β^2(G) \ge 1$ suffices for the existence of a $2$-factor. In this paper, we extend this result to general $k \ge 2$ by showing that any graph $G$ with even $k|V(G)|$ and $β^k(G) \ge 1$ contains a $k$-factor. Moreover, if $G$ is additionally a split graph of even order, then it admits a $(k+1)$-factor. We also prove that any graph $G$ with $β^k(G) \ge 1$ contains at least $k-1$ disjoint perfect or near-perfect matchings. Finally, for any bipartite graph $G$ with bipartition $(X, Y)$, we introduce an analogue of the $k$-th binding number and show that, under the condition $β^k(G, X) \ge 1$, the graph admits $k$ disjoint matchings, each covering $X$. |
| title | Strong binding numbers and factors |
| topic | Combinatorics 05C70 |
| url | https://arxiv.org/abs/2508.18555 |