Tight upper bounds on the hop domination number of triangle-free graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913721853411328 |
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| author | Fujita, Shinya Park, Boram |
| author_facet | Fujita, Shinya Park, Boram |
| contents | For a graph $G$, a subset $S$ of $V(G)$ is a {\it hop dominating set} of $G$ if every vertex not in $S$ has a $2$-step neighbor in $S$. The {\it hop domination number}, $γ_h(G)$, of $G$ is the minimum cardinality of a hop dominating set of $G$. In this paper, we show that for a connected triangle-free graph $G$ with $n\ge 15$ vertices, if $δ(G)\ge 2$, then $γ_h(G)\le \frac{2n}{5}$, and the bound is tight. We also give some tight upper bounds on $γ_h(G)$ for {triangle-free} graphs $G$ that contain a Hamiltonian path or a Hamiltonian cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04124 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tight upper bounds on the hop domination number of triangle-free graphs Fujita, Shinya Park, Boram Combinatorics For a graph $G$, a subset $S$ of $V(G)$ is a {\it hop dominating set} of $G$ if every vertex not in $S$ has a $2$-step neighbor in $S$. The {\it hop domination number}, $γ_h(G)$, of $G$ is the minimum cardinality of a hop dominating set of $G$. In this paper, we show that for a connected triangle-free graph $G$ with $n\ge 15$ vertices, if $δ(G)\ge 2$, then $γ_h(G)\le \frac{2n}{5}$, and the bound is tight. We also give some tight upper bounds on $γ_h(G)$ for {triangle-free} graphs $G$ that contain a Hamiltonian path or a Hamiltonian cycle. |
| title | Tight upper bounds on the hop domination number of triangle-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.04124 |