The γ–θ conjecture in eternal domination

Let $H$ be the Berlekamp–van Lint–Seidel graph on $243$ vertices, the Cayley graph of $\mathbb Z_3^5$ with strongly regular parameters $$(243,22,1,2),$$ and let $G=\overline H$. The proof establishes $$\gamma(G)=3$$ because every pair has a common neighbor in $H$, while an $H$-triangle gives a dominating triple in $G$. It then proves $$\gamma^\infty(G)=3$$ by showing that the family of all dominating triples is closed under a legal response to every attack: after moving one guard to the attacked vertex, another dominating triple can always be obtained. Finally, a double-counting argument shows that $H$ is not 3-colorable, hence $$\theta(G)=\chi(H)>3.$$ Therefore $$\gamma(G)=\gamma^\infty(G)=3<\theta(G).$$

Let $H$ be the Berlekamp–van Lint–Seidel graph on $243$ vertices, the Cayley graph of $\mathbb Z_3^5$ with strongly regular parameters $$(243,22,1,2),$$ and let $G=\overline H$. The proof establishes $$\gamma(G)=3$$ because every pair has a common neighbor in $H$, while an $H$-triangle gives a dominating triple in $G$. It then proves $$\gamma^\infty(G)=3$$ by showing that the family of all dominating triples is closed under a legal response to every attack: after moving one guard to the attacked vertex, another dominating triple can always be obtained. Finally, a double-counting argument shows that $H$ is not 3-colorable, hence $$\theta(G)=\chi(H)>3.$$ Therefore $$\gamma(G)=\gamma^\infty(G)=3<\theta(G).$$