On a conjecture of E\v{g}ecio\v{g}lu and Ir\v{s}i\v{c}

In 2021, {\"O}. E\v{g}ecio\v{g}lu, V. Ir\v{s}i\v{c} introduced the concept of Fibonacci-run graph $\mathcal{R}_{n}$ as an induced subgraph of Hypercube. They conjectured that the diameter of $\mathcal{R}_{n}$ is given by $n-\lfloor(1+\frac{n}{2})^{\frac{1}{2}}-\frac{3}{4}\rfloor$. In this paper, we introduce the novel concept of distance-barriers between vertices in $\mathcal{R}_{n}$ and provide an elegant method to give lower bound for the diameter of $\mathcal{R}_{n}$ via distance-barriers. By constructing different types of distance-barriers, we show that the conjecture does not hold for all $n\geq 230$ and some of $n$ between $91$ and $229$. Furthermore, lower bounds for the diameter of some Fibonacci-run graphs are obtained, which turn out to be better than the result given in the conjecture.