Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 4 2026 12 01 Bounding the Eviction Number of a Graph in Terms of its Independence Number 1221 1234 15166 10.22049/cco.2026.31031.2719 EN Gary MacGillivray Department of Mathematics and Statistics, University of Victoria, Victoria, Canada Christina M. Mynhardt Department of Mathematics and Statistics, University of Victoria, Victoria, Canada Virgelot Virgile Department of Mathematics and Statistics, University of Victoria, Victoria, Canada Journal Article 2025 09 24 An eternal dominating family of a graph $G$ in the eviction game is a collection $\mathcal{D}_{k}=\{D_{1},D_{2},\dots,D_{l}\}$ of dominating sets of $G$ such that (a) $|D_{i}|=|D_{j}|$ for all $i,j\in\{1,2,\dots,l\}$, and (b) for any $i\in \{1,2,\dots,l\}$ 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 $j\in\{1,2,\dots,l\}\setminus\{i\}$ 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.<br />We compare $e^{\infty}$ to the independence number $\alpha$. We show that the ratio $\alpha/e^{\infty}$ is unbounded and construct an infinite class of connected graphs for which $e^{\infty}/\alpha \approx 4/3$. 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 $f(k)$. https://comb-opt.azaruniv.ac.ir/article_15166_f0552c5dd4d26deaa32fdb880b67d435.pdf