Smaller Counterexamples to the Gamma-Theta Conjecture

Find graphs with fewer than 243 vertices satisfying γ(G) = γ∞(G) < θ(G), thereby providing smaller counterexamples to the Gamma-Theta conjecture for eternal dominating sets.

Background

The paper disproves the Gamma-Theta conjecture, which asserts that every graph G satisfying γ(G) = γ∞(G) also satisfies γ(G) = θ(G). The authors construct a 243-vertex graph G, the complement of the Berlekamp–van Lint–Seidel graph, and establish γ(G) = γ∞(G) = 3 while proving θ(G) ≥ 5. Consequently, G is a counterexample to the conjecture.

After presenting this counterexample, the authors explicitly leave unresolved the task of determining whether smaller graphs can exhibit the same strict inequality γ(G) = γ∞(G) < θ(G). They also suggest that potential smaller examples may possess substantial symmetry, but the concrete open problem is to find such graphs.