Isolation of non-triangle cycles in graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911200286081024 |
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| author | Borg, Peter Scicluna, Dayle |
| author_facet | Borg, Peter Scicluna, Dayle |
| contents | Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $ι(G, \mathcal{F})$, is the size of a smallest set $D$ of vertices of $G$ such that the closed neighbourhood of $D$ intersects the vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G-N[D]$ contains no $\mathcal{F}$-graph). Let $\mathcal{C}$ be the set of cycles, and let $\mathcal{C}'$ be the set of non-triangle cycles (that is, cycles of length at least $4$). Let $G$ be a connected graph having exactly $n$ vertices and $m$ edges. The first author proved that $ι(G,\mathcal{C}) \leq n/4$ if $G$ is not a triangle. Bartolo and the authors proved that $ι(G,\{C_4\}) \leq n/5$ if $G$ is not a copy of one of nine graphs. Various authors proved that $ι(G,\mathcal{C}) \leq (m+1)/5$ if $G$ is not a triangle. We prove that $ι(G,\mathcal{C}') \leq (m+1)/6$ if $G$ is not a $4$-cycle. Zhang and Wu established this for the case where $G$ is triangle-free. Our result yields the inequality $ι(G,\{C_4\}) \leq (m+1)/6$ of Wei, Zhang and Zhao. These bounds are attained by infinitely many (non-isomorphic) graphs. The proof of our inequality hinges on also determining the graphs attaining the bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isolation of non-triangle cycles in graphs Borg, Peter Scicluna, Dayle Combinatorics Discrete Mathematics 05C35, 05C38, 05C69 Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $ι(G, \mathcal{F})$, is the size of a smallest set $D$ of vertices of $G$ such that the closed neighbourhood of $D$ intersects the vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G-N[D]$ contains no $\mathcal{F}$-graph). Let $\mathcal{C}$ be the set of cycles, and let $\mathcal{C}'$ be the set of non-triangle cycles (that is, cycles of length at least $4$). Let $G$ be a connected graph having exactly $n$ vertices and $m$ edges. The first author proved that $ι(G,\mathcal{C}) \leq n/4$ if $G$ is not a triangle. Bartolo and the authors proved that $ι(G,\{C_4\}) \leq n/5$ if $G$ is not a copy of one of nine graphs. Various authors proved that $ι(G,\mathcal{C}) \leq (m+1)/5$ if $G$ is not a triangle. We prove that $ι(G,\mathcal{C}') \leq (m+1)/6$ if $G$ is not a $4$-cycle. Zhang and Wu established this for the case where $G$ is triangle-free. Our result yields the inequality $ι(G,\{C_4\}) \leq (m+1)/6$ of Wei, Zhang and Zhao. These bounds are attained by infinitely many (non-isomorphic) graphs. The proof of our inequality hinges on also determining the graphs attaining the bound. |
| title | Isolation of non-triangle cycles in graphs |
| topic | Combinatorics Discrete Mathematics 05C35, 05C38, 05C69 |
| url | https://arxiv.org/abs/2510.08361 |