Bounding the Eviction Number of a Graph in Terms of its Independence Number
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912602299301888 |
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| author | MacGillivray, Gary Mynhardt, Christina Virgile, Virgelot |
| author_facet | MacGillivray, Gary Mynhardt, Christina Virgile, Virgelot |
| contents | An eternal dominating family of graph $G$ in the eviction game is a collection $\mathcal{D}_{k}=\{D_{1},...,D_{l}\}$ of dominating sets of $G$ such that (a) $|D_{i}|=|D_{j}|$ for all $i,j\in\{1,2,...,l\}$, and (b) for any $i\in \{1,2,...,l\}$ and any $v\in D_{i}$, either all neighbours of $v$ belong to $D_{i}$, or there are a neighbour $w$ of $v$ not in $D_{i}$ and an integer $j\in\{1,2,...,l\}\setminus\{i\}$ such that $D_{i}\cup\{w\}\setminus \{v\}=D_{j}$. The eviction number of $G$, denoted by $e^{\infty}(G)$, is the smallest cardinality of the sets in such an eternal dominating family.
We compare $e^{\infty}$ to the independence number $α$. We show that the ratio $α/e^{\infty}$ is unbounded and construct an infinite class of connected graphs for which $e^{\infty}/α\approx 4/3$. As our main result, we use Ramsey numbers to show that for any integer $k\geq1$, there exists a function $f(k)$ such that any graph with independence number $k$ has eviction number at most $f(k)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounding the Eviction Number of a Graph in Terms of its Independence Number MacGillivray, Gary Mynhardt, Christina Virgile, Virgelot Combinatorics 05C69 An eternal dominating family of graph $G$ in the eviction game is a collection $\mathcal{D}_{k}=\{D_{1},...,D_{l}\}$ of dominating sets of $G$ such that (a) $|D_{i}|=|D_{j}|$ for all $i,j\in\{1,2,...,l\}$, and (b) for any $i\in \{1,2,...,l\}$ and any $v\in D_{i}$, either all neighbours of $v$ belong to $D_{i}$, or there are a neighbour $w$ of $v$ not in $D_{i}$ and an integer $j\in\{1,2,...,l\}\setminus\{i\}$ such that $D_{i}\cup\{w\}\setminus \{v\}=D_{j}$. The eviction number of $G$, denoted by $e^{\infty}(G)$, is the smallest cardinality of the sets in such an eternal dominating family. We compare $e^{\infty}$ to the independence number $α$. We show that the ratio $α/e^{\infty}$ is unbounded and construct an infinite class of connected graphs for which $e^{\infty}/α\approx 4/3$. As our main result, we use Ramsey numbers to show that for any integer $k\geq1$, there exists a function $f(k)$ such that any graph with independence number $k$ has eviction number at most $f(k)$. |
| title | Bounding the Eviction Number of a Graph in Terms of its Independence Number |
| topic | Combinatorics 05C69 |
| url | https://arxiv.org/abs/2509.19535 |