Bounding the Eviction Number of a Graph in Terms of its Independence Number

Abstract: 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 $\alpha$. 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 $f(k)$.