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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2509.09925 |
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| _version_ | 1866916947296256000 |
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| author | Huang, Jing |
| author_facet | Huang, Jing |
| contents | For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence number of $G$, denoted as $α_k(G)$, is the number of vertices in a maximum generalized $k$-independent set of $G$. For a graph $G$ with $n$ vertices, $m$ edges, $c$ connected components, and $c_1$ induced cycles of length 1 modulo 3, Bock et al. [J. Graph Theory 103 (2023) 661-673] showed that $α_3(G)\geq n-\frac{1}{3}(m+c+c_1)$ and identified the extremal graphs in which every two cycles are vertex-disjoint. Li and Zhou [Appl. Math. Comput. 484 (2025) 129018] proved that if $G$ is a tree with $n$ vertices, then $α_4(G) \geq \frac{3}{4}n$. They also presented all the corresponding extremal trees. In this paper, for a general graph $G$ with $n$ vertices, it is proved that $α_4(G)\geq \frac{3}{4}(n-ω(G))$ by using a different approach, where $ω(G)$ denotes the dimension of the cycle space of $G$. The graphs whose generalized $4$-independence number attains the lower bound are characterized completely. This represents a logical continuation of the work by Bock et al. and serves as a natural extension of the result by Li and Zhou. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09925 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A sharp lower bound on the generalized 4-independence number Huang, Jing Combinatorics For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence number of $G$, denoted as $α_k(G)$, is the number of vertices in a maximum generalized $k$-independent set of $G$. For a graph $G$ with $n$ vertices, $m$ edges, $c$ connected components, and $c_1$ induced cycles of length 1 modulo 3, Bock et al. [J. Graph Theory 103 (2023) 661-673] showed that $α_3(G)\geq n-\frac{1}{3}(m+c+c_1)$ and identified the extremal graphs in which every two cycles are vertex-disjoint. Li and Zhou [Appl. Math. Comput. 484 (2025) 129018] proved that if $G$ is a tree with $n$ vertices, then $α_4(G) \geq \frac{3}{4}n$. They also presented all the corresponding extremal trees. In this paper, for a general graph $G$ with $n$ vertices, it is proved that $α_4(G)\geq \frac{3}{4}(n-ω(G))$ by using a different approach, where $ω(G)$ denotes the dimension of the cycle space of $G$. The graphs whose generalized $4$-independence number attains the lower bound are characterized completely. This represents a logical continuation of the work by Bock et al. and serves as a natural extension of the result by Li and Zhou. |
| title | A sharp lower bound on the generalized 4-independence number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.09925 |